S
S
S
3 + 56 13 + 8 / 2
2 + 3 = 5
2 x = 6
x = 4
a x 2 + b x + c = 0
a   0
x = b ± b 2 4 a c 2 a
x 3 4 x 2 + 5 x 6
2 x = 6
x = 4
2 4
6   8
10 / 5 = 2
p
q
p
q
p
q
a x 2 + b x + c = 0
a   0
x = b ± b 2 4 a c 2 a
a x 2 + b x + c = 0
a   0
x = b ± b 2 4 a c 2 a
a x 2 + b x + c = 0
a   0
a x 2 + b x + c = 0 x 2 + b a x = c a x 2 + b a x + ( b 2 a ) 2 = ( b 2 a ) 2 c a ( x + b 2 a ) 2 = b 2 4 a c 4 a 2 x + b 2 a = ± b 2 4 a c 2 a x = b ± b 2 4 a c 2 a
r
s
r = s
p
q
q
p
x
x
A
X
a
A
a A
a
A
x
X = { x 1 , x 2 , , x n }
x 1 , x 2 , , x n
X = { x : x satisfies P }
x
X
P
E
E
E = { 2 , 4 , 6 , } or E = { x : x is an even integer and x > 0 }
2 E
E
3 E
3
E
N = { n : n is a natural number } = { 1 , 2 , 3 , } ; Z = { n : n is an integer } = { , 1 , 0 , 1 , 2 , } ; Q = { r : r is a rational number } = { p / q : p , q Z where q   0 } ; R = { x : x is a real number } ; C = { z : z is a complex number }
A
B
A B
B A
A
B
A
B
{ 4 , 5 , 8 } { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }
N Z Q R C
B
A
B A
B   A
A
B
A B
{ 4 , 7 , 9 } { 2 , 4 , 5 , 8 , 9 }
A = B
A B
B A
A B
A
B
A B = { x : x A or x B } ;
A
B
A B = { x : x A and x B }
A
B
A
B
A = { 1 , 3 , 5 }
B = { 1 , 2 , 3 , 9 }
A B = { 1 , 2 , 3 , 5 , 9 } and A B = { 1 , 3 }
i = 1 n A i = A 1 A n
i = 1 n A i = A 1 A n
A 1 , , A n
E
O
E
O
A
B
A B =
U
A U
A
A
A
A = { x : x U and x A }
A
B
A
B
A B = A B = { x : x A and x B }
R
A = { x R : 0 < x 3 } and B = { x R : 2 x < 4 }
A B = { x R : 2 x 3 } A B = { x R : 0 < x < 4 } A B = { x R : 0 < x < 2 } A = { x R : x 0 or x > 3 }
A
B
C
A A = A
A A = A
A A =
A = A
A =
A ( B C ) = ( A B ) C
A ( B C ) = ( A B ) C
A B = B A
A B = B A
A ( B C ) = ( A B ) ( A C )
A ( B C ) = ( A B ) ( A C )
A A = { x : x A or x A } = { x : x A } = A
A A = { x : x A and x A } = { x : x A } = A
A A = A A =
A
B
C
A ( B C ) = A { x : x B or x C } = { x : x A or x B , or x C } = { x : x A or x B } C = ( A B ) C
A ( B C ) = ( A B ) C
A
B
( A B ) = A B
( A B ) = A B
A B =
A
B
( A B ) A B
( A B ) A B
x ( A B )
x A B
x
A
B
x A
x B
x A B
( A B ) A B
x A B
x A
x B
x A
x B
x A B
x ( A B )
( A B ) A B
( A B ) = A B
( A B ) ( B A ) =
( A B ) ( B A ) = ( A B ) ( B A ) = A A B B =
A
B
A × B
A
B
A
B
A × B = { ( a , b ) : a A and b B }
A = { x , y }
B = { 1 , 2 , 3 }
C =
A × B
{ ( x , 1 ) , ( x , 2 ) , ( x , 3 ) , ( y , 1 ) , ( y , 2 ) , ( y , 3 ) }
A × C =
n
A 1 × × A n = { ( a 1 , , a n ) : a i A i for i = 1 , , n }
A = A 1 = A 2 = = A n
A n
A × × A
A
n
A × × A
n
R 3
A × B
f A × B
A
B
a A
b B
( a , b ) f
A
f
B
f : A B
A f B
( a , b ) A × B
f ( a ) = b
f : a b
A
f
f ( A ) = { f ( a ) : a A } B
f
A = { 1 , 2 , 3 }
B = { a , b , c }
f
g
A
B
f
g
1 A
B
g ( 1 ) = a
g ( 1 ) = b
f : A B
f : R R
f ( x ) = x 3
f : x x 3
f : Q Z
f ( p / q ) = p
1 / 2 = 2 / 4
f ( 1 / 2 ) = 1
2
f : A B
f
B
f ( A ) = B
f
a A
b B
f ( a ) = b
f
a 1   a 2
f ( a 1 )   f ( a 2 )
f ( a 1 ) = f ( a 2 )
a 1 = a 2
f : Z Q
f ( n ) = n / 1
f
g : Q Z
g ( p / q ) = p
p / q
g
f : A B
g : B C
f
g
A
C
( g f ) ( x ) = g ( f ( x ) )
f : A B
g : B C
g f : A C
f ( x ) = x 2
g ( x ) = 2 x + 5
( f g ) ( x ) = f ( g ( x ) ) = ( 2 x + 5 ) 2 = 4 x 2 + 20 x + 25
( g f ) ( x ) = g ( f ( x ) ) = 2 x 2 + 5
f g   g f
f g = g f
f ( x ) = x 3
g ( x ) = x 3
( f g ) ( x ) = f ( g ( x ) ) = f ( x 3 ) = ( x 3 ) 3 = x
( g f ) ( x ) = g ( f ( x ) ) = g ( x 3 ) = x 3 3 = x
2 × 2
A = ( a b c d )
T A : R 2 R 2
T A ( x , y ) = ( a x + b y , c x + d y )
( x , y )
R 2
( a b c d ) ( x y ) = ( a x + b y c x + d y )
R n
R m
S = { 1 , 2 , 3 }
π : S S
π ( 1 ) = 2 , π ( 2 ) = 1 , π ( 3 ) = 3
π
( 1 2 3 π ( 1 ) π ( 2 ) π ( 3 ) ) = ( 1 2 3 2 1 3 )
S
π : S S
S
f : A B
g : B C
h : C D
( h g ) f = h ( g f )
f
g
g f
f
g
g f
f
g
g f
h ( g f ) = ( h g ) f
a A
( h ( g f ) ) ( a ) = h ( ( g f ) ( a ) ) = h ( g ( f ( a ) ) ) = ( h g ) ( f ( a ) ) = ( ( h g ) f ) ( a )
f
g
c C
a A
( g f ) ( a ) = g ( f ( a ) ) = c
g
b B
g ( b ) = c
a A
f ( a ) = b
( g f ) ( a ) = g ( f ( a ) ) = g ( b ) = c
S
i d S
i d
S
i d ( s ) = s
s S
g : B A
f : A B
g f = i d A
f g = i d B
f 1
f
f
f ( x ) = x 3
f 1 ( x ) = x 3
f ( x ) = ln x
f 1 ( x ) = e x
f ( f 1 ( x ) ) = f ( e x ) = ln e x = x
f 1 ( f ( x ) ) = f 1 ( ln x ) = e ln x = x
A = ( 3 1 5 2 )
A
R 2
R 2
T A ( x , y ) = ( 3 x + y , 5 x + 2 y )
T A
A
T A 1 = T A 1
A 1 = ( 2 1 5 3 ) ;
T A 1 ( x , y ) = ( 2 x y , 5 x + 3 y )
T A 1 T A ( x , y ) = T A T A 1 ( x , y ) = ( x , y )
T B ( x , y ) = ( 3 x , 0 )
B = ( 3 0 0 0 )
T B 1 ( x , y ) = ( a x + b y , c x + d y )
( x , y ) = T B T B 1 ( x , y ) = ( 3 a x + 3 b y , 0 )
x
y
y
0
π = ( 1 2 3 2 3 1 )
S = { 1 , 2 , 3 }
π 1 = ( 1 2 3 3 1 2 )
π
f : A B
g : B A
g f = i d A
g ( f ( a ) ) = a
a 1 , a 2 A
f ( a 1 ) = f ( a 2 )
a 1 = g ( f ( a 1 ) ) = g ( f ( a 2 ) ) = a 2
f
b B
f
a A
f ( a ) = b
f ( g ( b ) ) = b
g ( b ) A
a = g ( b )
f
b B
f
a A
f ( a ) = b
f
a
g
g ( b ) = a
f
X
R X × X
( x , x ) R
x X
( x , y ) R
( y , x ) R
( x , y )
( y , z ) R
( x , z ) R
R
X
x y
( x , y ) R
=
p
q
r
s
q
s
p / q r / s
p s = q r
p / q r / s
r / s t / u
q
s
u
p s = q r
r u = s t
p s u = q r u = q s t
s   0
p u = q t
p / q t / u
f
g
R
f ( x ) g ( x )
f ( x ) = g ( x )
f ( x ) g ( x )
g ( x ) h ( x )
f ( x ) g ( x ) = c 1
g ( x ) h ( x ) = c 2
c 1
c 2
f ( x ) h ( x ) = ( f ( x ) g ( x ) ) + ( g ( x ) h ( x ) ) = c 1 + c 2
f ( x ) h ( x ) = 0
f ( x ) h ( x )
( x 1 , y 1 )
( x 2 , y 2 )
R 2
( x 1 , y 1 ) ( x 2 , y 2 )
x 1 2 + y 1 2 = x 2 2 + y 2 2
R 2
A
B
2 × 2
2 × 2
A B
P
P A P 1 = B
A = ( 1 2 1 1 ) and B = ( 18 33 11 20 )
A B
P A P 1 = B
P = ( 2 5 1 3 )
I
2 × 2
I = ( 1 0 0 1 )
I A I 1 = I A I = A
A B
P
P A P 1 = B
A = P 1 B P = P 1 B ( P 1 ) 1
A B
B C
P
Q
P A P 1 = B
Q B Q 1 = C
C = Q B Q 1 = Q P A P 1 Q 1 = ( Q P ) A ( Q P ) 1
P
X
X 1 , X 2 ,
X i X j =
i   j
k X k = X
X
x X
[ x ] = { y X : y x }
x
X
X
X
P = { X i }
X
X
X i
X
x X
x [ x ]
[ x ]
X = x X [ x ]
x , y X
[ x ] = [ y ]
[ x ] [ y ] =
[ x ]
[ y ]
z [ x ] [ y ]
z x
z y
x y
[ x ] [ y ]
[ y ] [ x ]
[ x ] = [ y ]
P = { X i }
X
x
y
y
x
x y
y x
x
y
y
z
x
z
( p , q )
( r , s )
f ( x )
g ( x )
R 2
( x 1 , y 1 ) ( x 2 , y 2 )
x 1 2 + y 1 2 = x 2 2 + y 2 2
r
s
n N
r
n
s
n
r
s
n
r s
n
r s = n k
k Z
r s ( mod n )
a
b
n
41 17 ( mod 8 )
41 17 = 24
8
n
Z
r
r r = 0
n
r s ( mod n )
r s = ( s r )
n
s r
n
s r ( mod n )
r s ( mod n )
s t ( mod n )
k
l
r s = k n
s t = l n
r t
n
r t = r s + s t = k n + l n = ( k + l ) n
r t
n
3
[ 0 ] = { , 3 , 0 , 3 , 6 , } , [ 1 ] = { , 2 , 1 , 4 , 7 , } , [ 2 ] = { , 1 , 2 , 5 , 8 , }
[ 0 ] [ 1 ] [ 2 ] = Z
[ 0 ]
[ 1 ]
[ 2 ]
n
n
A = { x : x N and x is even } , B = { x : x N and x is prime } , C = { x : x N and x is a multiple of 5 }
A B
B C
A B
A ( B C )
A B = { 2 }
B C = { 5 }
A = { a , b , c }
B = { 1 , 2 , 3 }
C = { x }
D =
A × B
B × A
A × B × C
A × D
A × B = { ( a , 1 ) , ( a , 2 ) , ( a , 3 ) , ( b , 1 ) , ( b , 2 ) , ( b , 3 ) , ( c , 1 ) , ( c , 2 ) , ( c , 3 ) }
A × D =
A
B
A × B = B × A
A = A
A =
A B = B A
A B = B A
A ( B C ) = ( A B ) ( A C )
x A ( B C )
x A
x B C
x A B
A C
x ( A B ) ( A C )
A ( B C ) ( A B ) ( A C )
x ( A B ) ( A C )
x A B
A C
x A
x
B
C
x A ( B C )
( A B ) ( A C ) A ( B C )
A ( B C ) = ( A B ) ( A C )
A ( B C ) = ( A B ) ( A C )
A B
A B = A
( A B ) = A B
A B = ( A B ) ( A B ) ( B A )
( A B ) ( A B ) ( B A ) = ( A B ) ( A B ) ( B A ) = [ A ( B B ) ] ( B A ) = A ( B A ) = ( A B ) ( A A ) = A B
( A B ) × C = ( A × C ) ( B × C )
( A B ) B =
( A B ) B = A B
A ( B C ) = ( A B ) ( A C )
A ( B C ) = A ( B C ) = ( A A ) ( B C ) = ( A B ) ( A C ) = ( A B ) ( A C )
A ( B C ) = ( A B ) ( A C )
( A B ) ( B A ) = ( A B ) ( A B )
f : Q Q
f
f ( p / q ) = p + 1 p 2
f ( p / q ) = 3 p 3 q
f ( p / q ) = p + q q 2
f ( p / q ) = 3 p 2 7 q 2 p q
f ( 2 / 3 )
f ( 1 / 2 ) = 3 / 4
f ( 2 / 4 ) = 3 / 8
f : R R
f ( x ) = e x
f : Z Z
f ( n ) = n 2 + 3
f : R R
f ( x ) = sin x
f : Z Z
f ( x ) = x 2
f
f ( R ) = { x R : x > 0 }
f
f ( R ) = { x : 1 x 1 }
f : A B
g : B C
f 1
g 1
( g f ) 1 = f 1 g 1
f : N N
f : N N
f ( n ) = n + 1
R 2
( x 1 , y 1 ) ( x 2 , y 2 )
x 1 2 + y 1 2 = x 2 2 + y 2 2
f : A B
g : B C
f
g
g f
g f
g
g f
f
g f
f
g
g f
g
f
x , y A
g ( f ( x ) ) = ( g f ) ( x ) = ( g f ) ( y ) = g ( f ( y ) )
f ( x ) = f ( y )
x = y
g f
c C
c = ( g f ) ( x ) = g ( f ( x ) )
x A
f ( x ) B
g
f ( x ) = x + 1 x 1
f
f
f f 1
f 1 f
f 1 ( x ) = ( x + 1 ) / ( x 1 )
f : X Y
A 1 , A 2 X
B 1 , B 2 Y
f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 )
f ( A 1 A 2 ) f ( A 1 ) f ( A 2 )
f 1 ( B 1 B 2 ) = f 1 ( B 1 ) f 1 ( B 2 )
f 1 ( B ) = { x X : f ( x ) B }
f 1 ( B 1 B 2 ) = f 1 ( B 1 ) f 1 ( B 2 )
f 1 ( Y B 1 ) = X f 1 ( B 1 )
y f ( A 1 A 2 )
x A 1 A 2
f ( x ) = y
y f ( A 1 )
f ( A 2 )
y f ( A 1 ) f ( A 2 )
f ( A 1 A 2 ) f ( A 1 ) f ( A 2 )
y f ( A 1 ) f ( A 2 )
y f ( A 1 )
f ( A 2 )
x
A 1
A 2
f ( x ) = y
x A 1 A 2
f ( x ) = y
f ( A 1 ) f ( A 2 ) f ( A 1 A 2 )
f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 )
x y
R
x y
m n
Z
m n > 0
x y
R
| x y | 4
m n
Z
m n ( mod 6 )
0
R 2
( a , b ) ( c , d )
a 2 + b 2 c 2 + d 2
m × n
R n
R m
x y
y x
x x
X = N { 2 }
x y
x + y N
R 2 { ( 0 , 0 ) }
( x 1 , y 1 ) ( x 2 , y 2 )
λ
( x 1 , y 1 ) = ( λ x 2 , λ y 2 )
R 2 ( 0 , 0 )
P ( R )
300 !
10
66
46
3 1 = 2
1 + 2 + + n = n ( n + 1 ) 2
n
n = 1
2
3
4
n
( n + 1 )
n = 1
1 = 1 ( 1 + 1 ) 2
n
1 + 2 + + n + ( n + 1 ) = n ( n + 1 ) 2 + n + 1 = n 2 + 3 n + 2 2 = ( n + 1 ) [ ( n + 1 ) + 1 ] 2
( n + 1 )
S
N
S
S ( n )
n N
S ( n 0 )
n 0
k
k n 0
S ( k )
S ( k + 1 )
S ( n )
n
n 0
n 3
2 n > n + 4
8 = 2 3 > 3 + 4 = 7
n 0 = 3
2 k > k + 4
k 3
2 k + 1 = 2 2 k > 2 ( k + 4 )
2 ( k + 4 ) = 2 k + 8 > k + 5 = ( k + 1 ) + 4
k
n 3
10 n + 1 + 3 10 n + 5
9
n N
n = 1
10 1 + 1 + 3 10 + 5 = 135 = 9 15
9
10 k + 1 + 3 10 k + 5
9
k 1
10 ( k + 1 ) + 1 + 3 10 k + 1 + 5 = 10 k + 2 + 3 10 k + 1 + 50 45 = 10 ( 10 k + 1 + 3 10 k + 5 ) 45
9
( a + b ) n = k = 0 n ( n k ) a k b n k
a
b
n N
( n k ) = n ! k ! ( n k ) !
n
n ! / ( k ! ( n k ) ! )
( n + 1 k ) = ( n k ) + ( n k 1 )
( n k ) + ( n k 1 ) = n ! k ! ( n k ) ! + n ! ( k 1 ) ! ( n k + 1 ) ! = ( n + 1 ) ! k ! ( n + 1 k ) ! = ( n + 1 k )
n = 1
n
1
( a + b ) n + 1 = ( a + b ) ( a + b ) n = ( a + b ) ( k = 0 n ( n k ) a k b n k ) = k = 0 n ( n k ) a k + 1 b n k + k = 0 n ( n k ) a k b n + 1 k = a n + 1 + k = 1 n ( n k 1 ) a k b n + 1 k + k = 1 n ( n k ) a k b n + 1 k + b n + 1 = a n + 1 + k = 1 n [ ( n k 1 ) + ( n k ) ] a k b n + 1 k + b n + 1 = k = 0 n + 1 ( n + 1 k ) a k b n + 1 k
S ( n )
n N
S ( n 0 )
n 0
S ( n 0 ) , S ( n 0 + 1 ) , , S ( k )
S ( k + 1 )
k n 0
S ( n )
n n 0
S
Z
S
Z
1
S = { n N : n 1 }
1 S
n S
0 < 1
n = n + 0 < n + 1
1 n < n + 1
n S
n + 1
S
S = N
N
S
S
S
S
k
1 k n
S
S
n + 1
S
S
n + 1
n + 1
S
S
S
n
S
n !
n
n ! = 1 2 3 ( n 1 ) n
1 ! = 1
n ! = n ( n 1 ) !
n > 1
a
b
b > 0
q
r
a = b q + r
0 r < b
q
r
q
r
q = q
r = r
q
r
S = { a b k : k Z and a b k 0 }
0 S
b
a
q = a / b
r = 0
0 S
S
a > 0
a b 0 S
a < 0
a b ( 2 a ) = a ( 1 2 b ) S
S  
S
r = a b q
a = b q + r
r 0
r < b
r > b
a b ( q + 1 ) = a b q b = r b > 0
a b ( q + 1 )
S
a b ( q + 1 ) < a b q
r = a b q
S
r b
0 S
r   b
r < b
q
r
r
r
q
q
a = b q + r , 0 r < b and a = b q + r , 0 r < b
b q + r = b q + r
r r
b ( q q ) = r r
b
r r
0 r r r < b
r r = 0
r = r
q = q
a
b
b = a k
k
a b
d
a
b
d a
d b
a
b
d
d
a
b
d
a
b
d d
a
b
a
b
d = gcd ( a , b )
gcd ( 24 , 36 ) = 12
gcd ( 120 , 102 ) = 6
a
b
gcd ( a , b ) = 1
a
b
r
s
gcd ( a , b ) = a r + b s
a
b
S = { a m + b n : m , n Z and a m + b n > 0 }
S
S
d = a r + b s
d = gcd ( a , b )
a = d q + r
0 r < d
r > 0
r = a d q = a ( a r + b s ) q = a a r q b s q = a ( 1 r q ) + b ( s q )
S
d
S
r = 0
d
a
d
b
d
a
b
d
a
b
d d
a = d h
b = d k
d = a r + b s = d h r + d k s = d ( h r + k s )
d
d
d
a
b
a
b
r
s
a r + b s = 1
945
2415
2415 = 945 2 + 525 945 = 525 1 + 420 525 = 420 1 + 105 420 = 105 4 + 0
105
420
105
525
105
945
105
2415
105
945
2415
d
945
2415
d
105
gcd ( 945 , 2415 ) = 105
r
s
945 r + 2415 s = 105
105 = 525 + ( 1 ) 420 = 525 + ( 1 ) [ 945 + ( 1 ) 525 ] = 2 525 + ( 1 ) 945 = 2 [ 2415 + ( 2 ) 945 ] + ( 1 ) 945 = 2 2415 + ( 5 ) 945
r = 5
s = 2
r
s
r = 41
s = 16
gcd ( a , b ) = d
r 1 > r 2 > > r n = d
b = a q 1 + r 1 a = r 1 q 2 + r 2 r 1 = r 2 q 3 + r 3 r n 2 = r n 1 q n + r n r n 1 = r n q n + 1
r
s
a r + b s = d
d = r n = r n 2 r n 1 q n = r n 2 q n ( r n 3 q n 1 r n 2 ) = q n r n 3 + ( 1 + q n q n 1 ) r n 2 = r a + s b
d
a
b
d
a
b
p
p > 1
p
p
p
1
p
n > 1
a
b
p
p a b
p a
p b
p
a
p b
gcd ( a , p ) = 1
r
s
a r + p s = 1
b = b ( a r + p s ) = ( a b ) r + p ( b s )
p
a b
p
b = ( a b ) r + p ( b s )
p 1 , p 2 , , p n
P = p 1 p 2 p n + 1
P
p i
1 i n
p i
P p 1 p 2 p n = 1
P
p   p i
P
n
n > 1
n = p 1 p 2 p k
p 1 , , p k
n = q 1 q 2 q l
k = l
q i
p i
n
n = 2
n
m
1 m < n
n = p 1 p 2 p k = q 1 q 2 q l
p 1 p 2 p k
q 1 q 2 q l
p 1 q i
i = 1 , , l
q 1 p j
j = 1 , , k
p i
q i
p 1 = q i
q 1 = p j
p 1 = q 1
p 1 p j = q 1 q i = p 1
n = p 2 p k = q 2 q l
k = l
q i = p i
i = 1 , , k
S
S
a
a
a
1
a
a = a 1 a 2
1 < a 1 < a
1 < a 2 < a
a 1 S
a 2 S
a
S
a 1 = p 1 p r a 2 = q 1 q s
a = a 1 a 2 = p 1 p r q 1 q s
a S
n
f
f ( n )
n
2 2 n + 1
n
2 2 5 + 1 = 4,294,967,297
2
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
4 × 10 18
123456792
84
52
r
s
r ( 84 ) + s ( 52 ) = gcd ( 84 , 52 )
1 2 + 2 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6
n N
S ( 1 ) : [ 1 ( 1 + 1 ) ( 2 ( 1 ) + 1 ) ] / 6 = 1 = 1 2
S ( k ) : 1 2 + 2 2 + + k 2 = [ k ( k + 1 ) ( 2 k + 1 ) ] / 6
1 2 + 2 2 + + k 2 + ( k + 1 ) 2 = [ k ( k + 1 ) ( 2 k + 1 ) ] / 6 + ( k + 1 ) 2 = [ ( k + 1 ) ( ( k + 1 ) + 1 ) ( 2 ( k + 1 ) + 1 ) ] / 6
S ( k + 1 )
S ( n )
n
1 3 + 2 3 + + n 3 = n 2 ( n + 1 ) 2 4
n N
n ! > 2 n
n 4
S ( 4 ) : 4 ! = 24 > 16 = 2 4
S ( k ) : k ! > 2 k
( k + 1 ) ! = k ! ( k + 1 ) > 2 k 2 = 2 k + 1
S ( k + 1 )
S ( n )
n
x + 4 x + 7 x + + ( 3 n 2 ) x = n ( 3 n 1 ) x 2
n N
10 n + 1 + 10 n + 1
3
n N
4 10 2 n + 9 10 2 n 1 + 5
99
n N
a 1 a 2 a n n 1 n k = 1 n a k
f ( n ) ( x )
f ( n )
n
f
( f g ) ( n ) ( x ) = k = 0 n ( n k ) f ( k ) ( x ) g ( n k ) ( x )
1 + 2 + 2 2 + + 2 n = 2 n + 1 1
n N
1 2 + 1 6 + + 1 n ( n + 1 ) = n n + 1
n N
x
( 1 + x ) n 1 n x
n = 0 , 1 , 2 ,
S ( 0 ) : ( 1 + x ) 0 1 = 0 0 = 0 x
S ( k ) : ( 1 + x ) k 1 k x
( 1 + x ) k + 1 1 = ( 1 + x ) ( 1 + x ) k 1 = ( 1 + x ) k + x ( 1 + x ) k 1 k x + x ( 1 + x ) k k x + x = ( k + 1 ) x
S ( k + 1 )
S ( n )
n
X
X
P ( X )
X
X
P ( { a , b } ) = { , { a } , { b } , { a , b } }
n
n
2 n
S N
1 S
n + 1 S
n S
S = N
a
b
gcd ( a , b )
r
s
gcd ( a , b ) = r a + s b
14
39
234
165
1739
9923
471
562
23771
19945
4357
3754
a
b
r
s
a r + b s = 1
a
b
1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 ,
f 1 = 1
f 2 = 1
f n + 2 = f n + 1 + f n
n N
f n < 2 n
f n + 1 f n 1 = f n 2 + ( 1 ) n
n 2
f n = [ ( 1 + 5 ) n ( 1 5 ) n ] / 2 n 5
lim n f n / f n + 1 = ( 5 1 ) / 2
f n
f n + 1
f 1 = 1
f 2 = 1
f n + 2 = f n + 1 + f n
a
b
gcd ( a , b ) = 1
r
s
a r + b s = 1
gcd ( a , s ) = gcd ( r , b ) = gcd ( r , s ) = 1
x , y N
x y
x
y
4 k
4 k + 1
k
a , b , r , s
a 2 + b 2 = r 2 a 2 b 2 = s 2
a
r
s
b
n N
n
0 , 1 , , n 1
r
s
Z
0 s < n
[ r ] = [ s ]
n
a
b
lcm ( a , b )
m
a
b
m
a
b
n
m
n
m
n
a
b
d = gcd ( a , b )
m = lcm ( a , b )
d m = | a b |
lcm ( a , b ) = a b
gcd ( a , b ) = 1
gcd ( a , c ) = gcd ( b , c ) = 1
gcd ( a b , c ) = 1
a
b
c
a , b , c Z
gcd ( a , b ) = 1
a b c
a c
gcd ( a , b ) = 1
r
s
a r + b s = 1
a c r + b c s = c
p 2
2 p 1
p
6 n + 5
2
3
6 n + 1
6 n + 5
6 k + 5
4 n 1
2
p
q
p 2 = 2 q 2
2
N
n
1 < n < N
2
3
5
4
N
N
N = 250
N
N
N = 120
N
N 0 = N { 0 }
A : N 0 × N 0 N 0
A ( 0 , y ) = y + 1 , A ( x + 1 , 0 ) = A ( x , 1 ) , A ( x + 1 , y + 1 ) = A ( x , A ( x + 1 , y ) )
A ( 3 , 1 )
A ( 4 , 1 )
A ( 5 , 1 )
a
b
gcd ( a , b )
r
s
gcd ( a , b ) = r a + s b
a
b
r
0 r < b
a = b q + r
q
( a r ) / b
q
b
a
a
b
a
b
a
b
a
b
r
s
r a + s b = gcd ( a , b )
2
a
a
b 1
2600 = 2 3 × 5 2 × 13
2600
1
1
c = 4 598 037 234
d = 7
d = 11
Z
2 × 2
2 × 2
n
n
a
b
n
n
a b
n
Z
n
Z n
n
12
[ 0 ] = { , 12 , 0 , 12 , 24 , } , [ 1 ] = { , 11 , 1 , 13 , 25 , } , [ 11 ] = { , 1 , 11 , 23 , 35 , }
0 , 1 , , 11
[ 0 ] , [ 1 ] , , [ 11 ]
Z n
a
b
n
( a + b ) ( mod n )
a + b
n
n
( a b ) ( mod n )
a b
n
n
7 + 4 1 ( mod 5 ) 7 3 1 ( mod 5 ) 3 + 5 0 ( mod 8 ) 3 5 7 ( mod 8 ) 3 + 4 7 ( mod 12 ) 3 4 0 ( mod 12 )
n
0
n
Z n
Z 8
2
4
6
n = 2
4
6
k
k n 1 ( mod 8 )
Z 8
0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 0 2 4 6 0 2 4 6 3 0 3 6 1 4 7 2 5 4 0 4 0 4 0 4 0 4 5 0 5 2 7 4 1 6 3 6 0 6 4 2 0 6 4 2 7 0 7 6 5 4 3 2 1
Z n
n
a , b , c Z n
a + b b + a ( mod n ) a b b a ( mod n )
( a + b ) + c a + ( b + c ) ( mod n ) ( a b ) c a ( b c ) ( mod n )
a + 0 a ( mod n ) a 1 a ( mod n )
a ( b + c ) a b + a c ( mod n )
a
a
a + ( a ) 0 ( mod n )
a
gcd ( a , n ) = 1
b
a ( mod n )
b
a b 1 ( mod n )
n
a + b
n
b + a
n
gcd ( a , n ) = 1
r
s
a r + n s = 1
n s = 1 a r
a r 1 ( mod n )
b
r
a b 1 ( mod n )
b
a b 1 ( mod n )
n
a b 1
k
a b n k = 1
d = gcd ( a , n )
d
a b n k
d
1
d = 1
180
360
90
A B C
A B C
A
B
C
S
π : S S
3 ! = 6
3 2 1 = 3 ! = 6
A
B
B
C
C
A
( A B C B C A )
120
A B C
μ 1 ρ 1
ρ 1
μ 1
( μ 1 ρ 1 ) ( A ) = μ 1 ( ρ 1 ( A ) ) = μ 1 ( B ) = C ( μ 1 ρ 1 ) ( B ) = μ 1 ( ρ 1 ( B ) ) = μ 1 ( C ) = B ( μ 1 ρ 1 ) ( C ) = μ 1 ( ρ 1 ( C ) ) = μ 1 ( A ) = A
μ 2
ρ 1
μ 1
μ 3
ρ 1 μ 1   μ 1 ρ 1
A B C
α
β
α β = i d
i d ρ 1 ρ 2 μ 1 μ 2 μ 3 i d i d ρ 1 ρ 2 μ 1 μ 2 μ 3 ρ 1 ρ 1 ρ 2 i d μ 3 μ 1 μ 2 ρ 2 ρ 2 i d ρ 1 μ 2 μ 3 μ 1 μ 1 μ 1 μ 2 μ 3 i d ρ 1 ρ 2 μ 2 μ 2 μ 3 μ 1 ρ 2 i d ρ 1 μ 3 μ 3 μ 1 μ 2 ρ 1 ρ 2 i d
n
G
G × G G
( a , b ) G × G
a b
a b
G
a
b
( G , )
G
( a , b ) a b
( a b ) c = a ( b c )
a , b , c G
e G
a G
e a = a e = a
a G
a 1
a a 1 = a 1 a = e
G
a b = b a
a , b G
Z = { , 1 , 0 , 1 , 2 , }
m , n Z
+
m + n
m n
0
n Z
n
n 1
m + n = n + m
a b
a b
m + n
n
m n
m + ( n )
n
n
Z 5
0
1
2
3
4
Z 5
m + n
Z 5
2 + 3 = 3 + 2 = 0
Z 5
Z n = { 0 , 1 , , n 1 }
n
( Z 5 , + )
+ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3
Z n
Z n
1 k = k 1 = k
k Z n
0
0 k = k 0 = 0
k
Z n
Z n { 0 }
2 Z 6
0 2 = 0 1 2 = 2 2 2 = 4 3 2 = 0 4 2 = 2 5 2 = 4
k
Z n
k
n
Z n
U ( n )
Z n
U ( n )
Z n
U ( 8 )
U ( 8 )
1 3 5 7 1 1 3 5 7 3 3 1 7 5 5 5 7 1 3 7 7 5 3 1
α β = β α
α
β
S 3
D 3
M 2 ( R )
2 × 2
G L 2 ( R )
M 2 ( R )
n × n
R
A = ( a b c d )
G L 2 ( R )
A 1
A A 1 = A 1 A = I
I
2 × 2
A
A
det A = a d b c   0
A
I = ( 1 0 0 1 )
A G L 2 ( R )
A 1 = 1 a d b c ( d b c a )
A B = B A
G L 2 ( R )
1 = ( 1 0 0 1 ) I = ( 0 1 1 0 ) J = ( 0 i i 0 ) K = ( i 0 0 i )
i 2 = 1
I 2 = J 2 = K 2 = 1
I J = K
J K = I
K I = J
J I = K
K J = I
I K = J
Q 8 = { ± 1 , ± I , ± J , ± K }
Q 8
C
C
1
z = a + b i
z 1 = a b i a 2 + b 2
z
G
n
| G | = n
Z 5
5
Z
| Z | =
G
e G
e g = g e = g
g G
e
e
G
e g = g e = g
e g = g e = g
g G
e = e
e
e e = e
e
e e = e
e = e e = e
g
g
g
G
g g = g g = e
g g = g g = e
g = g
g = g e = g ( g g ) = ( g g ) g = e g = g
g
G
g
g 1
G
a , b G
( a b ) 1 = b 1 a 1
a , b G
a b b 1 a 1 = a e a 1 = a a 1 = e
b 1 a 1 a b = e
( a b ) 1 = b 1 a 1
G
a G
( a 1 ) 1 = a
a 1 ( a 1 ) 1 = e
a
( a 1 ) 1 = e ( a 1 ) 1 = a a 1 ( a 1 ) 1 = a e = a
a
b
G
x G
a x = b
x
G
a
b
G
a x = b
x a = b
G
a x = b
x
a x = b
a 1
x = e x = a 1 a x = a 1 b
x 1
x 2
a x = b
a x 1 = b = a x 2
x 1 = a 1 a x 1 = a 1 a x 2 = x 2
x a = b
G
a , b , c G
b a = c a
b = c
a b = a c
b = c
G
g G
g 0 = e
n N
g n = g g g n times
g n = g 1 g 1 g 1 n times
g , h G
g m g n = g m + n
m , n Z
( g m ) n = g m n
m , n Z
( g h ) n = ( h 1 g 1 ) n
n Z
G
( g h ) n = g n h n
( g h ) n   g n h n
Z
Z n
n g
g n
m g + n g = ( m + n ) g
m , n Z
m ( n g ) = ( m n ) g
m , n Z
m ( g + h ) = m g + m h
n Z
Z
Z n
2 Z = { , 2 , 0 , 2 , 4 , }
H
G
H
G
G
H
H
G
H = { e }
G
G
R
1
a R
1 / a
Q = { p / q : p and q are nonzero integers }
R
R
1
1 = 1 / 1
R
Q
Q
p / q
r / s
p r / q s
Q
p / q Q
Q
( p / q ) 1 = q / p
R
Q
C
H = { 1 , 1 , i , i }
H
C
H
H C
S L 2 ( R )
G L 2 ( R )
A = ( a b c d )
S L 2 ( R )
a d b c = 1
S L 2 ( R )
2 × 2
S L 2 ( R )
A
A 1 = ( d b c a )
S L 2 ( R )
H
G
G
H
G
G
2 × 2
M 2 ( R )
2 × 2
M 2 ( R )
M 2 ( R )
( 1 0 0 1 ) + ( 1 0 0 1 ) = ( 0 0 0 0 )
G L 2 ( R )
Z 4
0
2
Z 2
Z 2 × Z 2
( a , b ) + ( c , d ) = ( a + c , b + d )
Z 2 × Z 2
Z 2 × Z 2
H 1 = { ( 0 , 0 ) , ( 0 , 1 ) }
H 2 = { ( 0 , 0 ) , ( 1 , 0 ) }
H 3 = { ( 0 , 0 ) , ( 1 , 1 ) }
Z 4
Z 2 × Z 2
Z 2 × Z 2
+ ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 1 ) ( 0 , 1 ) ( 0 , 0 ) ( 1 , 1 ) ( 1 , 0 ) ( 1 , 0 ) ( 1 , 0 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 1 ) ( 1 , 1 ) ( 1 , 0 ) ( 0 , 1 ) ( 0 , 0 )
H
G
e
G
H
h 1 , h 2 H
h 1 h 2 H
h H
h 1 H
H
G
H
e H
e H = e
e
G
e H e H = e H
e e H = e H e = e H
e e H = e H e H
e = e H
H
h H
H
h H
h h = h h = e
G
h = h 1
H
G
H
G
H
G
H  
g , h H
g h 1
H
H
G
g h 1 H
g
h
H
h
H
h 1
H
g h 1 H
H G
H  
g h 1 H
g , h H
g H
g g 1 = e
H
g H
e g 1 = g 1
H
h 1 , h 2 H
H
h 1 ( h 2 1 ) 1 = h 1 h 2 H
H
G
Z 8
6 + 7
2 1
U ( 16 )
5 7
3 1
x Z
3 x 2 ( mod 7 )
5 x + 1 13 ( mod 23 )
5 x + 1 13 ( mod 26 )
9 x 3 ( mod 5 )
5 x 1 ( mod 6 )
3 x 1 ( mod 6 )
3 + 7 Z = { , 4 , 3 , 10 , }
18 + 26 Z
5 + 6 Z
G = { a , b , c , d }
a b c d a a c d a b b b c d c c d a b d d a b c
a b c d a a b c d b b a d c c c d a b d d c b a
a b c d a a b c d b b c d a c c d a b d d a b c
a b c d a a b c d b b a c d c c b a d d d d b c
( Z 4 , + )
D 4
U ( 12 )
1 5 7 11 1 1 5 7 11 5 5 1 11 7 7 7 11 1 5 11 11 7 5 1
S = R { 1 }
S
a b = a + b + a b
( S , )
A
B
G L 2 ( R )
A B   B A
S L 2 ( R )
( 1 x y 0 1 z 0 0 1 )
( 1 x y 0 1 z 0 0 1 ) ( 1 x y 0 1 z 0 0 1 ) = ( 1 x + x y + y + x z 0 1 z + z 0 0 1 )
det ( A B ) = det ( A ) det ( B )
G L 2 ( R )
G L 2 ( R )
A
B
G L 2 ( R )
A B G L 2 ( R )
Z 2 n = { ( a 1 , a 2 , , a n ) : a i Z 2 }
Z 2 n
( a 1 , a 2 , , a n ) + ( b 1 , b 2 , , b n ) = ( a 1 + b 1 , a 2 + b 2 , , a n + b n )
Z 2 n
R = R { 0 }
R
Z
G = R × Z
G
( a , m ) ( b , n ) = ( a b , m + n )
G
G
g , h G
( g h ) n   g n h n
n !
n
σ = ( 1 2 n a 1 a 2 a n )
S n
a i
n
a 1
n 1
a 2 ,
a n 1
a n
σ
n ( n 1 ) 2 1 = n !
0 + a a + 0 a ( mod n )
a Z n
n
a 1 a ( mod n )
a Z n
b Z n
a + b b + a 0 ( mod n )
n
n
n
n
a ( b + c ) a b + a c ( mod n )
a
b
G
a b n a 1 = ( a b a 1 ) n
n Z
( a b a 1 ) n = ( a b a 1 ) ( a b a 1 ) ( a b a 1 ) = a b ( a a 1 ) b ( a a 1 ) b b ( a a 1 ) b a 1 = a b n a 1
U ( n )
Z n
n > 2
k U ( n )
k 2 = 1
k   1
g 1 g 2 g n
g n 1 g n 1 1 g 1 1
G
a , b G
x a = b
G
G
G
b a = c a
b = c
a b = a c
b = c
a , b , c G
a 2 = e
a
G
G
a b a b = ( a b ) 2 = e = a 2 b 2 = a a b b
b a = a b
G
a G
a
a 2 = e
G
( a b ) 2 = a 2 b 2
a
b
G
G
Z 3 × Z 3
Z 3 × Z 3
Z 9
H 1 = { i d }
H 2 = { i d , ρ 1 , ρ 2 }
H 3 = { i d , μ 1 }
H 4 = { i d , μ 2 }
H 5 = { i d , μ 3 }
S 3
H = { 2 k : k Z }
H
Q
n = 0 , 1 , 2 ,
n Z = { n k : k Z }
n Z
Z
Z
T = { z C : | z | = 1 }
T
C
G
2 × 2
( cos θ sin θ sin θ cos θ )
θ R
G
S L 2 ( R )
G = { a + b 2 : a , b Q and a and b are not both zero }
R
G
1 = 1 + 0 2
( a + b 2 ) ( c + d 2 ) = ( a c + 2 b d ) + ( a d + b c ) 2
G
( a + b 2 ) 1 = a / ( a 2 2 b 2 ) b 2 / ( a 2 2 b 2 )
G
2 × 2
H = { ( a b c d ) : a + d = 0 }
H
G
S L 2 ( Z )
2 × 2
S L 2 ( R )
Q 8
G
G
H
K
G
H K
G
S 3
H
K
G
H K = { h k : h H and k K }
G
G
G
g G
Z ( G ) = { x G : g x = x g for all g G }
G
G
a
b
G
a 4 b = b a
a 3 = e
a b = b a
b a = a 4 b = a 3 a b = a b
x y = x 1 y 1
x
y
G
G
H
G
C ( H ) = { g G : g h = h g for all h H }
C ( H )
G
H
G
H
G
g G
g H g 1 = { g h g 1 : h H }
G
d 1 d 2 d 12
3 d 1 + 1 d 2 + 3 d 3 + + 3 d 11 + 1 d 12 0 ( mod 10 )
d 12
( d 1 , d 2 , , d k ) ( w 1 , w 2 , , w k ) 0 ( mod n )
d 1 w 1 + d 2 w 2 + + d k w k 0 ( mod n )
( d 1 , d 2 , , d k ) ( w 1 , w 2 , , w k ) 0 ( mod n )
k
d 1 d 2 d k
0 d i < n
gcd ( w i , n ) = 1
1 i k
( d 1 , d 2 , , d k ) ( w 1 , w 2 , , w k ) 0 ( mod n )
k
d 1 d 2 d k
0 d i < n
d i
d j
gcd ( w i w j , n ) = 1
i
j
1
k
( d 1 , d 2 , , d 10 ) ( 10 , 9 , , 1 ) 0 ( mod 11 )
d 10
6
6 1
6.00000
6.00000 + 0.00000 i
6
8
n
n
ρ 2
ρ 2 = ( A B C C A B ) = ( 1 2 3 3 1 2 )
f g
( f g ) ( x ) = f ( g ( x ) )
g
f g
f
μ
ρ
1
I
J
K
1
1
1 1 = 1
I
I
i = 1
S 8
4
4
Z
Z n
3 Z
3
3 Z = { , 3 , 0 , 3 , 6 , }
3 Z
3
3
3
H = { 2 n : n Z }
H
Q
a = 2 m
b = 2 n
H
a b 1 = 2 m 2 n = 2 m n
H
H
Q
2
G
a
G
a
a = { a k : k Z }
G
a
G
a
a
a 0 = e
g
h
a
a
g = a m
h = a n
m
n
g h = a m a n = a m + n
a
g = a n
a
g 1 = a n
a
H
G
a
a
H
a
a
G
a
a = { n a : n Z }
a G
a
a
G
a
G = a
G
a
G
a
G
a
n
a n = e
| a | = n
a
n
a
| a | =
a
1
5
Z 6
Z 6
2 Z 6
3
2
2 = { 0 , 2 , 4 }
Z
Z n
1
1
Z
Z n
Z n
Z 6
U ( 9 )
Z 9
U ( 9 )
{ 1 , 2 , 4 , 5 , 7 , 8 }
U ( 9 )
2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 7 2 5 = 5 2 6 = 1
S 3
S 3
S 3
G
a G
G
g
h
G
a
g = a r
h = a s
g h = a r a s = a r + s = a s + r = a s a r = h g
G
G
G
G
G
G
a
H
G
H = { e }
H
H
g
g
a n
n
H
g 1 = a n
H
n
n
H
a
n > 0
m
a m H
m
h = a m
H
h H
h
h H
H
G
h = a k
k
q
r
k = m q + r
0 r < m
a k = a m q + r = ( a m ) q a r = h q a r
a r = a k h q
a k
h q
H
a r
H
m
a m
H
r = 0
k = m q
h = a k = a m q = h q
H
h
Z
n Z
n = 0 , 1 , 2 ,
G
n
a
G
a k = e
n
k
a k = e
k = n q + r
0 r < n
e = a k = a n q + r = a n q a r = e a r = a r
m
a m = e
n
r = 0
n
k
k = n s
s
a k = a n s = ( a n ) s = e s = e
G
n
a G
b = a k
b
n / d
d = gcd ( k , n )
m
e = b m = a k m
m
n
k m
n / d
m ( k / d )
d
n
k
n / d
k / d
n / d
m ( k / d )
m
m
n / d
Z n
r
1 r < n
gcd ( r , n ) = 1
Z 16
1
3
5
7
9
11
13
15
Z 16
16
Z 16
1 9 = 9 2 9 = 2 3 9 = 11 4 9 = 4 5 9 = 13 6 9 = 6 7 9 = 15 8 9 = 8 9 9 = 1 10 9 = 10 11 9 = 3 12 9 = 12 13 9 = 5 14 9 = 14 15 9 = 7
C = { a + b i : a , b R }
i 2 = 1
z = a + b i
a
z
b
z
z = a + b i
w = c + d i
z + w = ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i
i 2 = 1
z
w
( a + b i ) ( c + d i ) = a c + b d i 2 + a d i + b c i = ( a c b d ) + ( a d + b c ) i
z = a + b i
z 1 C
z z 1 = z 1 z = 1
z = a + b i
z 1 = a b i a 2 + b 2
z = a + b i
z ¯ = a b i
z = a + b i
| z | = a 2 + b 2
z = 2 + 3 i
w = 1 2 i
z + w = ( 2 + 3 i ) + ( 1 2 i ) = 3 + i
z w = ( 2 + 3 i ) ( 1 2 i ) = 8 i
z 1 = 2 13 3 13 i | z | = 13 z ¯ = 2 3 i
z = a + b i
x y
a
x
b
y
z 1 = 2 + 3 i
z 2 = 1 2 i
z 3 = 3 + 2 i
θ
x
r
z = a + b i = r ( cos θ + i sin θ )
r = | z | = a 2 + b 2
a = r cos θ b = r sin θ
r ( cos θ + i sin θ )
r cis θ
cos θ + i sin θ
z
0 θ < 360
0 θ < 2 π
z = 2 cis 60
a = 2 cos 60 = 1
b = 2 sin 60 = 3
z = 1 + 3 i
z = 3 2 3 2 i
r = a 2 + b 2 = 36 = 6
θ = arctan ( b a ) = arctan ( 1 ) = 315
3 2 3 2 i = 6 cis 315
z = r cis θ
w = s cis ϕ
z w = r s cis ( θ + ϕ )
z = 3 cis ( π / 3 )
w = 2 cis ( π / 6 )
z w = 6 cis ( π / 2 ) = 6 i
z = r cis θ
[ r cis θ ] n = r n cis ( n θ )
n = 1 , 2 ,
n
n = 1
k
1 k n
z n + 1 = z n z = r n ( cos n θ + i sin n θ ) r ( cos θ + i sin θ ) = r n + 1 [ ( cos n θ cos θ sin n θ sin θ ) + i ( sin n θ cos θ + cos n θ sin θ ) ] = r n + 1 [ cos ( n θ + θ ) + i sin ( n θ + θ ) ] = r n + 1 [ cos ( n + 1 ) θ + i sin ( n + 1 ) θ ]
z = 1 + i
z 10
( 1 + i ) 10
z 10
z 10 = ( 1 + i ) 10 = ( 2 cis ( π 4 ) ) 10 = ( 2 ) 10 cis ( 5 π 2 ) = 32 cis ( π 2 ) = 32 i
C
Q
R
C
T = { z C : | z | = 1 }
C
H = { 1 , 1 , i , i }
H
1
1
i
i
z 4 = 1
z n = 1
n
n
z n = 1
n
z = cis ( 2 k π n )
k = 0 , 1 , , n 1
n
T
n
z n = cis ( n 2 k π n ) = cis ( 2 k π ) = 1
z
2 k π / n
2 π
z n = 1
n
n
n
T
n
n
n
ω = 2 2 + 2 2 i ω 3 = 2 2 + 2 2 i ω 5 = 2 2 2 2 i ω 7 = 2 2 2 2 i
2 2
2 8
2 2 1,000,000
2 37,398,332 ( mod 46,389 )
0
46,388
n
a
2
a = 2 k 1 + 2 k 2 + + 2 k n
k 1 < k 2 < < k n
a
57 = 2 0 + 2 3 + 2 4 + 2 5
Z n
b a x ( mod n )
c a y ( mod n )
b c a x + y ( mod n )
a 2 k ( mod n )
k
a 2 0 ( mod n ) a 2 1 ( mod n ) a 2 k ( mod n )
n
271 321 ( mod 481 )
321 = 2 0 + 2 6 + 2 8 ;
271 321 ( mod 481 )
271 2 0 + 2 6 + 2 8 271 2 0 271 2 6 271 2 8 ( mod 481 )
271 2 i ( mod 481 )
i = 0 , 6 , 8
271 2 1 = 73,441 329 ( mod 481 )
271 2 2 ( mod 481 )
271 2 2 ( 271 2 1 ) 2 ( mod 481 ) ( 329 ) 2 ( mod 481 ) 108,241 ( mod 481 ) 16 ( mod 481 )
( a 2 n ) 2 a 2 2 n a 2 n + 1 ( mod n )
271 2 6 419 ( mod 481 )
271 2 8 16 ( mod 481 )
271 321 271 2 0 + 2 6 + 2 8 ( mod 481 ) 271 2 0 271 2 6 271 2 8 ( mod 481 ) 271 419 16 ( mod 481 ) 1,816,784 ( mod 481 ) 47 ( mod 481 )
n
3
U ( 20 )
5
U ( 23 )
Z 8
5 th
15 40 ( mod 23 )
Z 60
U ( 8 )
Q
G
G
5 Z 12
3 R
3 R
i C
72 Z 240
312 Z 471
12
10
Z
7
Z 24
15
Z 12
Z 60
Z 13
Z 48
U ( 20 )
U ( 18 )
R
7
C
i
i 2 = 1
C
2 i
C
( 1 + i ) / 2
C
( 1 + 3 i ) / 2
7 Z = { , 7 , 0 , 7 , 14 , }
{ 0 , 3 , 6 , 9 , 12 , 15 , 18 , 21 }
{ 0 }
{ 0 , 6 }
{ 0 , 4 , 8 }
{ 0 , 3 , 6 , 9 }
{ 0 , 2 , 4 , 6 , 8 , 10 }
{ 1 , 3 , 7 , 9 }
{ 1 , 1 , i , i }
G L 2 ( R )
( 0 1 1 0 )
( 0 1 / 3 3 0 )
( 1 1 1 0 )
( 1 1 0 1 )
( 1 1 1 0 )
( 3 / 2 1 / 2 1 / 2 3 / 2 )
( 1 0 0 1 ) , ( 1 0 0 1 ) , ( 0 1 1 0 ) , ( 0 1 1 0 )
( 1 0 0 1 ) , ( 1 1 1 0 ) , ( 1 1 1 0 ) , ( 0 1 1 1 ) , ( 0 1 1 1 ) , ( 1 0 0 1 )
Z 18
D 4
Q 8
U ( 30 )
Z 32
Z
Q
R
0
1 , 1
a 24 = e
G
a
1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
n
n 20
U ( n )
A = ( 0 1 1 0 ) and B = ( 0 1 1 1 )
G L 2 ( R )
A
B
A B
( 3 2 i ) + ( 5 i 6 )
( 4 5 i ) ( 4 i 4 ) ¯
( 5 4 i ) ( 7 + 2 i )
( 9 i ) ( 9 i ) ¯
i 45
( 1 + i ) + ( 1 + i ) ¯
3 + 3 i
43 18 i
i
a + b i
2 cis ( π / 6 )
5 cis ( 9 π / 4 )
3 cis ( π )
cis ( 7 π / 4 ) / 2
3 + i
3
1 i
5
2 + 2 i
3 + i
3 i
2 i + 2 3
2 cis ( 7 π / 4 )
2 2 cis ( π / 4 )
3 cis ( 3 π / 2 )
( 1 + i ) 1
( 1 i ) 6
( 3 + i ) 5
( i ) 10
( ( 1 i ) / 2 ) 4
( 2 2 i ) 12
( 2 + 2 i ) 5
( 1 i ) / 2
16 ( i 3 )
1 / 4
| z | = | z ¯ |
z z ¯ = | z | 2
z 1 = z ¯ / | z | 2
| z + w | | z | + | w |
| z w | | | z | | w | |
| z w | = | z | | w |
292 3171 ( mod 582 )
2557 341 ( mod 5681 )
2071 9521 ( mod 4724 )
971 321 ( mod 765 )
292
1523
a , b G
a
a 1
g G
| a | = | g 1 a g |
a b
b a
p
q
Z p q
p
r
Z p r
Z p
p
g
h
15
16
G
g h
| g h | = 1
a
G
a m a n
Z n
n > 2
G
a
b G
| a | = m
| b | = n
gcd ( m , n ) = 1
a b = { e }
G
G
G
g , h G
m
n
( g 1 ) m = e
( g h ) m n = e
G
G
G
n
x
y = x k
gcd ( k , n ) = 1
y
G
G
2
G
4
G
p q
gcd ( p , q ) = 1
G
a
b
p
q
G
Z
n Z
n = 0 , 1 , 2 ,
Z n
r
1 r < n
gcd ( r , n ) = 1
G
G
g
G
g
G
g
G
G
G
m
d m
G
d
n
1
n
z = r ( cos θ + i sin θ )
w = s ( cos ϕ + i sin ϕ )
z w = r s [ cos ( θ + ϕ ) + i sin ( θ + ϕ ) ]
C
n
T
n
α T
α m = 1
α n = 1
α d = 1
d = gcd ( m , n )
z C
| z |   1
z
z = cos θ + i sin θ
T
θ Q
z
2
a x ( mod n )
n
x
Z
3 Z
Z n
Z 14
1
12
1
12
12
T
n
14
r
C
1
θ
2 π 14
n
U ( n )
40
40
U ( 40 )
7
U
U
7
7
U ( 40 )
U ( 49 )
U ( 49 )
U ( 35 )
16
U ( 35 )
U ( n )
n
A B C
S = { A , B , C }
π : S S
( A B C A B C ) ( A B C C A B ) ( A B C B C A ) ( A B C A C B ) ( A B C C B A ) ( A B C B A C )
( A B C B C A )
A
B
B
C
C
A
A B B C C A
X
S X
X
X = { 1 , 2 , , n }
S n
S X
n
S n
n
n
S n
n !
S n
1
1
2
2
n
n
f : S n S n
f 1
f
| S n | = n !
S n
G
S 5
i d
σ = ( 1 2 3 4 5 1 2 3 5 4 ) τ = ( 1 2 3 4 5 3 2 1 4 5 ) μ = ( 1 2 3 4 5 3 2 1 5 4 )
G
i d σ τ μ i d i d σ τ μ σ σ i d μ τ τ τ μ i d σ μ μ τ σ i d
σ
τ
X
σ
τ
( σ τ ) ( x ) = σ ( τ ( x ) )
τ
σ
σ τ
τ
σ
σ τ ( x )
σ ( τ ( x ) )
σ ( x )
( x ) σ
σ = ( 1 2 3 4 4 1 2 3 ) τ = ( 1 2 3 4 2 1 4 3 )
σ τ = ( 1 2 3 4 1 4 3 2 )
τ σ = ( 1 2 3 4 3 2 1 4 )
σ S X
k
a 1 , a 2 , , a k X
σ ( a 1 ) = a 2 σ ( a 2 ) = a 3 σ ( a k ) = a 1
σ ( x ) = x
x X
( a 1 , a 2 , , a k )
σ
k
σ = ( 1 2 3 4 5 6 7 6 3 5 1 4 2 7 ) = ( 1 6 2 3 5 4 )
6
τ = ( 1 2 3 4 5 6 1 4 2 3 5 6 ) = ( 2 4 3 )
3
( 1 2 3 4 5 6 2 4 1 3 6 5 ) = ( 1 2 4 3 ) ( 5 6 )
4
σ = ( 1 3 5 2 ) and τ = ( 2 5 6 )
σ
1 3 , 3 5 , 5 2 , 2 1
τ
2 5 , 5 6 , 6 2
σ τ
τ
σ
1 3 , 3 5 , 5 6 , 6 2 1
σ τ = ( 1 3 5 6 )
μ = ( 1634 )
σ μ = ( 1 6 5 2 ) ( 3 4 )
S X
σ = ( a 1 , a 2 , , a k )
τ = ( b 1 , b 2 , , b l )
a i   b j
i
j
( 1 3 5 )
( 2 7 )
( 1 3 5 )
( 3 4 7 )
( 1 3 5 ) ( 2 7 ) = ( 1 3 5 ) ( 2 7 ) ( 1 3 5 ) ( 3 4 7 ) = ( 1 3 4 7 5 )
σ
τ
S X
σ τ = τ σ
σ = ( a 1 , a 2 , , a k )
τ = ( b 1 , b 2 , , b l )
σ τ ( x ) = τ σ ( x )
x X
x
{ a 1 , a 2 , , a k }
{ b 1 , b 2 , , b l }
σ
τ
x
σ ( x ) = x
τ ( x ) = x
σ τ ( x ) = σ ( τ ( x ) ) = σ ( x ) = x = τ ( x ) = τ ( σ ( x ) ) = τ σ ( x )
x { a 1 , a 2 , , a k }
σ ( a i ) = a ( i mod k ) + 1
a 1 a 2 a 2 a 3 a k 1 a k a k a 1
τ ( a i ) = a i
σ
τ
σ τ ( a i ) = σ ( τ ( a i ) ) = σ ( a i ) = a ( i mod k ) + 1 = τ ( a ( i mod k ) + 1 ) = τ ( σ ( a i ) ) = τ σ ( a i )
x { b 1 , b 2 , , b l }
σ
τ
S n
X = { 1 , 2 , , n }
σ S n
X 1
{ σ ( 1 ) , σ 2 ( 1 ) , }
X 1
X
i
X
X 1
X 2
{ σ ( i ) , σ 2 ( i ) , }
X 2
X 3 , X 4 ,
X
r
σ i
σ i ( x ) = { σ ( x ) x X i x x X i
σ = σ 1 σ 2 σ r
X 1 , X 2 , , X r
σ 1 , σ 2 , , σ r
σ = ( 1 2 3 4 5 6 6 4 3 1 5 2 ) τ = ( 1 2 3 4 5 6 3 2 1 5 6 4 )
σ = ( 1624 ) τ = ( 13 ) ( 456 ) σ τ = ( 1 3 6 ) ( 2 4 5 ) τ σ = ( 1 4 3 ) ( 2 5 6 )
( 1 )
2
( a 1 , a 2 , , a n ) = ( a 1 a n ) ( a 1 a n 1 ) ( a 1 a 3 ) ( a 1 a 2 )
( 1 6 ) ( 2 5 3 ) = ( 1 6 ) ( 2 3 ) ( 2 5 ) = ( 1 6 ) ( 4 5 ) ( 2 3 ) ( 4 5 ) ( 2 5 )
( 1 2 ) ( 1 2 )
( 1 3 ) ( 2 4 ) ( 1 3 ) ( 2 4 )
( 1 6 )
( 2 3 ) ( 1 6 ) ( 2 3 )
( 3 5 ) ( 1 6 ) ( 1 3 ) ( 1 6 ) ( 1 3 ) ( 3 5 ) ( 5 6 )
( 1 6 )
r
i d = τ 1 τ 2 τ r
r
r
r > 1
r = 2
r > 2
τ r 1 τ r
( a b ) ( a b ) = i d ( b c ) ( a b ) = ( a c ) ( b c ) ( c d ) ( a b ) = ( a b ) ( c d ) ( a c ) ( a b ) = ( a b ) ( b c )
a
b
c
d
τ r 1 τ r
i d = τ 1 τ 2 τ r 3 τ r 2
r 2
r
τ r 1 τ r
r
a
τ r 2 τ r 1
r 2
r
a
τ r 2
r 2
τ r 3 τ r 2
a
a
r 2
σ
σ
σ
σ
σ = σ 1 σ 2 σ m = τ 1 τ 2 τ n
m
n
σ
σ m σ 1
i d = σ σ m σ 1 = τ 1 τ n σ m σ 1
n
σ
S n
A n
n
A n
n
A n
S n
A n
A n
σ
σ = σ 1 σ 2 σ r
σ i
r
σ 1 = σ r σ r 1 σ 1
A n
S n
n 2
A n
n ! / 2
A n
S n
B n
σ
S n
n 2
σ
λ σ : A n B n
λ σ ( τ ) = σ τ
λ σ ( τ ) = λ σ ( μ )
σ τ = σ μ
τ = σ 1 σ τ = σ 1 σ μ = μ
λ σ
λ σ
A 4
S 4
A 4
( 1 ) ( 12 ) ( 34 ) ( 13 ) ( 24 ) ( 14 ) ( 23 ) ( 123 ) ( 132 ) ( 124 ) ( 142 ) ( 134 ) ( 143 ) ( 234 ) ( 243 )
A 4
n
n
n = 3 , 4 ,
n
D n
n
1 , 2 , , n
n
k
k + 1
k 1
2 n
n
n
D n
S n
2 n
D n
n 3
r
s
r n = 1 s 2 = 1 s r s = r 1
n
n
i d , 360 n , 2 360 n , , ( n 1 ) 360 n
360 / n
r
r
r k = k 360 n
n
n
s 1 , s 2 , , s n
s k
k
n
s 1 = s n / 2 + 1 , s 2 = s n / 2 + 2 , , s n / 2 = s n
s 1 , s 2 , , s n
s k
s = s 1
s 2 = 1
r n = 1
t
n
k
k + 1
k 1
k + 1
t = r k
k 1
t = r k s
r
s
D n
D n
r
s
D n = { 1 , r , r 2 , , r n 1 , s , r s , r 2 s , , r n 1 s }
s r s = r 1
n
D 4
1
2
3
4
r = ( 1234 ) r 2 = ( 13 ) ( 24 ) r 3 = ( 1432 ) r 4 = ( 1 )
s 1 = ( 24 ) s 2 = ( 13 )
D 4
8
r s 1 = ( 12 ) ( 34 ) r 3 s 1 = ( 14 ) ( 23 )
D 4
n
6
6 4 = 24
24
S 4
24
S 4
1
2
3
4
S 4
180
S 4
S 4
S 4
( 1 3 4 ) ( 3 5 4 )
A 3
( 1 2 3 4 5 2 4 1 5 3 )
( 1 2 3 4 5 4 2 5 1 3 )
( 1 2 3 4 5 3 5 1 4 2 )
( 1 2 3 4 5 1 4 3 2 5 )
( 12453 )
( 13 ) ( 25 )
( 1345 ) ( 234 )
( 12 ) ( 1253 )
( 143 ) ( 23 ) ( 24 )
( 1423 ) ( 34 ) ( 56 ) ( 1324 )
( 1254 ) ( 13 ) ( 25 )
( 1254 ) ( 13 ) ( 25 ) 2
( 1254 ) 1 ( 123 ) ( 45 ) ( 1254 )
( 1254 ) 2 ( 123 ) ( 45 )
( 123 ) ( 45 ) ( 1254 ) 2
( 1254 ) 100
| ( 1254 ) |
| ( 1254 ) 2 |
( 12 ) 1
( 12537 ) 1
[ ( 12 ) ( 34 ) ( 12 ) ( 47 ) ] 1
[ ( 1235 ) ( 467 ) ] 1
( 135 ) ( 24 )
( 14 ) ( 23 )
( 1324 )
( 134 ) ( 25 )
( 17352 )
( 14356 )
( 156 ) ( 234 )
( 1426 ) ( 142 )
( 17254 ) ( 1423 ) ( 154632 )
( 142637 )
( 16 ) ( 15 ) ( 13 ) ( 14 )
( 16 ) ( 14 ) ( 12 )
( a 1 , a 2 , , a n ) 1
( a 1 , a 2 , , a n ) 1 = ( a 1 , a n , a n 1 , , a 2 )
S 4
{ σ S 4 : σ ( 1 ) = 3 }
{ σ S 4 : σ ( 2 ) = 2 }
{ σ S 4 : σ ( 1 ) = 3
σ ( 2 ) = 2 }
S 4
{ ( 13 ) , ( 13 ) ( 24 ) , ( 132 ) , ( 134 ) , ( 1324 ) , ( 1342 ) }
A 4
S 7
A 7
A 10
15
( 12345 ) ( 678 )
A 8
26
S n
n = 3 , , 10
A 5
A 6
( 1 ) , ( a 1 , a 2 ) ( a 3 , a 4 ) , ( a 1 , a 2 , a 3 ) , ( a 1 , a 2 , a 3 , a 4 , a 5 )
A 5
σ S n
n
i
j
σ i = σ j
i j ( mod n )
σ = σ 1 σ m S n
σ
σ 1 , , σ m
D 5
r
s
r
s
( 12 ) ( 34 )
A 4
S n
n 3
( 123 ) ( 12 )
( 12 ) ( 123 )
A n
n 4
D n
n 3
σ S n
σ
n 1
σ S n
σ
σ
n 2
σ
σ
σ
σ 2
3
A n
n 3
3
( a b ) ( b c )
( a b ) ( c d )
S n
( 1 2 ) , ( 13 ) , , ( 1 n )
( 1 2 ) , ( 23 ) , , ( n 1 , n )
( 12 ) , ( 1 2 n )
G
λ g : G G
λ g ( a ) = g a
λ g
G
n !
n
G
Z ( G ) = { g G : g x = x g for all x G }
D 8
D 10
D n
τ = ( a 1 , a 2 , , a k )
k
σ
σ τ σ 1 = ( σ ( a 1 ) , σ ( a 2 ) , , σ ( a k ) )
k
μ
k
σ
σ τ σ 1 = μ
σ τ σ 1 ( σ ( a i ) ) = σ ( a i + 1 )
α
β
S n
α β
σ S n
σ α σ 1 = β
S n
σ S X
σ n ( x ) = y
n Z
x y
X
x X
σ S X
O x , σ = { y : x y }
{ 1 , 2 , 3 , 4 , 5 }
S 5
α = ( 1254 ) β = ( 123 ) ( 45 ) γ = ( 13 ) ( 25 )
O x , σ O y , σ  
O x , σ = O y , σ
σ
H
S X
x , y X
σ H
σ ( x ) = y
σ
O x , σ = X
x X
α S n
n 3
α β = β α
β S n
α
S n
α
α 1
α
σ A n
τ S n
τ 1 σ τ A n
α 1 β 1 α β
α , β S n
r
s
D n
s r s = r 1
r k s = s r k
D n
r k D n
n / gcd ( k , n )
1
n
S 4
σ
σ
τ σ
n
n !
n
2 n
n
n
n
n ! / 2
S 4
1
4
1
5
8
5
1
6
2
24
S 10
a 3
b c
a d 1 b
a , b , c , d
G
K
K
L
G
L
L
A 4
A 4
A 4
24
S 8
1
6
S 6
S 10
2
30
N
G
H
G
H
g G
g H = { g h : h H }
H g = { h g : h H }
H
Z 6
0
3
0 + H = 3 + H = { 0 , 3 } 1 + H = 4 + H = { 1 , 4 } 2 + H = 5 + H = { 2 , 5 }
Z
Z n
H
S 3
{ ( 1 ) , ( 123 ) , ( 132 ) }
H
( 1 ) H = ( 1 2 3 ) H = ( 132 ) H = { ( 1 ) , ( 1 23 ) , ( 132 ) } ( 1 2 ) H = ( 1 3 ) H = ( 2 3 ) H = { ( 1 2 ) , ( 1 3 ) , ( 2 3 ) }
H
H ( 1 ) = H ( 1 2 3 ) = H ( 132 ) = { ( 1 ) , ( 1 23 ) , ( 132 ) } H ( 1 2 ) = H ( 1 3 ) = H ( 2 3 ) = { ( 1 2 ) , ( 1 3 ) , ( 2 3 ) }
K
S 3
{ ( 1 ) , ( 1 2 ) }
K
( 1 ) K = ( 1 2 ) K = { ( 1 ) , ( 1 2 ) } ( 1 3 ) K = ( 1 2 3 ) K = { ( 1 3 ) , ( 1 2 3 ) } ( 2 3 ) K = ( 1 3 2 ) K = { ( 2 3 ) , ( 1 3 2 ) } ;
K
K ( 1 ) = K ( 1 2 ) = { ( 1 ) , ( 1 2 ) } K ( 1 3 ) = K ( 1 3 2 ) = { ( 1 3 ) , ( 1 3 2 ) } K ( 2 3 ) = K ( 1 2 3 ) = { ( 2 3 ) , ( 1 2 3 ) }
H
G
g 1 , g 2 G
g 1 H = g 2 H
H g 1 1 = H g 2 1
g 1 H g 2 H
g 2 g 1 H
g 1 1 g 2 H
H
G
H
G
H
G
G
G
H
G
g 1 H
g 2 H
H
G
g 1 H g 2 H =
g 1 H = g 2 H
g 1 H g 2 H  
a g 1 H g 2 H
a = g 1 h 1 = g 2 h 2
h 1
h 2
H
g 1 = g 2 h 2 h 1 1
g 1 g 2 H
g 1 H = g 2 H
G
H
G
H
G
H
G
H
G
[ G : H ]
H
G
G = Z 6
H = { 0 , 3 }
[ G : H ] = 3
G = S 3
H = { ( 1 ) , ( 123 ) , ( 132 ) }
K = { ( 1 ) , ( 12 ) }
[ G : H ] = 2
[ G : K ] = 3
H
G
H
G
H
G
L H
R H
H
G
H
G
H
G
ϕ : L H R H
g H L H
ϕ ( g H ) = H g 1
ϕ
g 1 H = g 2 H
H g 1 1 = H g 2 1
ϕ
H g 1 1 = ϕ ( g 1 H ) = ϕ ( g 2 H ) = H g 2 1
g 1 H = g 2 H
ϕ
ϕ ( g 1 H ) = H g
H
G
g G
ϕ : H g H
ϕ ( h ) = g h
ϕ
H
g H
ϕ
ϕ ( h 1 ) = ϕ ( h 2 )
h 1 , h 2 H
h 1 = h 2
ϕ ( h 1 ) = g h 1
ϕ ( h 2 ) = g h 2
g h 1 = g h 2
h 1 = h 2
ϕ
g H
g h
h H
ϕ ( h ) = g h
G
H
G
| G | / | H | = [ G : H ]
H
G
H
G
G
[ G : H ]
| H |
| G | = [ G : H ] | H |
G
g G
g
G
| G | = p
p
G
g G
g   e
g
G
g   e
g
| g | > 1
p
g
G
p
Z p
H
K
G
G H K
[ G : K ] = [ G : H ] [ H : K ]
[ G : K ] = | G | | K | = | G | | H | | H | | K | = [ G : H ] [ H : K ]
A 4
12
6
12
1
2
3
4
6
A 4
6
H
A 4
3
H
3
H
3
6
A 4
6
[ A 4 : H ] = 2
H
A 4
H
g H = H g
g H g 1 = H
g A 4
3
A 4
3
H
( 123 )
H
( 123 ) 1 = ( 132 )
H
g h g 1 H
g A 4
h H
( 124 ) ( 123 ) ( 124 ) 1 = ( 124 ) ( 123 ) ( 142 ) = ( 243 ) ( 243 ) ( 123 ) ( 243 ) 1 = ( 243 ) ( 123 ) ( 234 ) = ( 142 )
H
( 1 ) , ( 123 ) , ( 132 ) , ( 243 ) , ( 243 ) 1 = ( 234 ) , ( 142 ) , ( 142 ) 1 = ( 124 )
A 4
6
τ
μ
S n
σ S n
μ = σ τ σ 1
τ = ( a 1 , a 2 , , a k ) μ = ( b 1 , b 2 , , b k )
σ
σ ( a 1 ) = b 1 σ ( a 2 ) = b 2 σ ( a k ) = b k
μ = σ τ σ 1
τ = ( a 1 , a 2 , , a k )
k
σ S n
σ ( a i ) = b
σ ( a ( i mod k ) + 1 ) = b
μ ( b ) = b
μ = ( σ ( a 1 ) , σ ( a 2 ) , , σ ( a k ) )
σ
μ
τ
ϕ
ϕ
ϕ : N N
ϕ ( n ) = 1
n = 1
n > 1
ϕ ( n )
m
1 m < n
gcd ( m , n ) = 1
U ( n )
Z n
ϕ ( n )
| U ( 12 ) | = ϕ ( 12 ) = 4
p
ϕ ( p ) = p 1
U ( n )
Z n
| U ( n ) | = ϕ ( n )
a
n
n > 0
gcd ( a , n ) = 1
a ϕ ( n ) 1 ( mod n )
U ( n )
ϕ ( n )
a ϕ ( n ) = 1
a U ( n )
a ϕ ( n ) 1
n
a ϕ ( n ) 1 ( mod n )
n = p
ϕ ( p ) = p 1
p
p a
p
a
a
b
a p 1 1 ( mod p )
b
b p b ( mod p )
3
Z 9
{ ( ) , ( 1 2 ) ( 3 4 ) , ( 1 3 ) ( 2 4 ) , ( 1 4 ) ( 2 3 ) }
S 4
S 4
G
G
p = 137909
57 137909 ( mod 137909 )
G
g
5
h
7
| G | 35
g
h
G
G
60
G
60
8
Z 24
3
U ( 8 )
3 Z
Z
A 4
S 4
A n
S n
D 4
S 4
T
C
H = { ( 1 ) , ( 123 ) , ( 132 ) }
S 4
8
1 + 8
2 + 8
3 + 8
4 + 8
5 + 8
6 + 8
7 + 8
3 Z
1 + 3 Z
2 + 3 Z
S L 2 ( R )
G L 2 ( R )
S L 2 ( R )
G L 2 ( R )
n = 15
a = 4
4 ϕ ( 15 ) 4 8 1 ( mod 15 )
p = 4 n + 3
x 2 1 ( mod p )
H
G
g 1 , g 2 G
g 1 H = g 2 H
H g 1 1 = H g 2 1
g 1 H g 2 H
g 2 g 1 H
g 1 1 g 2 H
g h g 1 H
g G
h H
g H = H g
g G
g 1 g H
g 1 H g
g H H g
ϕ : L H R H
ϕ ( g H ) = H g
g n = e
g
n
σ
σ
σ = ( 12 ) ( 345 ) ( 78 ) ( 9 )
( 2 , 3 , 2 , 1 )
( 1 , 2 , 2 , 3 )
α , β S n
γ
β = γ α γ 1
β = γ α γ 1
γ S n
α
β
| G | = 2 n
2
G
[ G : H ] = 2
a
b
H
a b H
[ G : H ] = 2
g H = H g
H
K
G
g H g K
H K
G
g ( H K ) = g H g K
H
K
G
G
a b
h H
k K
h a k = b
H = { ( 1 ) , ( 123 ) , ( 132 ) }
A 4
G
n
ϕ ( n )
G
n = p 1 e 1 p 2 e 2 p k e k
p 1 , p 2 , , p k
ϕ ( n ) = n ( 1 1 p 1 ) ( 1 1 p 2 ) ( 1 1 p k )
gcd ( m , n ) = 1
ϕ ( m n ) = ϕ ( m ) ϕ ( n )
n = d n ϕ ( d )
n
A 4
A 4
12
A 4
S 4
S 4
S 4
ϕ
G
m
m
G
G
m
A 4
G
m
391 = 17 23
a
b
p
100
1000
n
n 1
n
100
1000
0 < n < 100
1 a n
7
S 7
7 ! = 5040
10
1
2
5040
n
A
B
C
B
C
A
B
C
A
B
C
C
C
f
f 1
f
f 1
f
A = 00 , B = 01 , , Z = 25
f ( p ) = p + 3 mod 26 ;
A D , B E , , Z C
f 1 ( p ) = p 3 mod 26 = p + 23 mod 26
3 , 14 , 9 , 7 , 4 , 20 , 3
0 , 11 , 6 , 4 , 1 , 17 , 0
3
26
26
b
f ( p ) = p + b mod 26
E = 04
S = 18
18 = 4 + b mod 26
b = 14
f ( p ) = p + 14 mod 26
f 1 ( p ) = p + 12 mod 26
26
f ( p ) = a p + b mod 26
f 1
c = a p + b mod 26
p
a
gcd ( a , 26 ) = 1
f 1 ( p ) = a 1 p a 1 b mod 26
f ( p ) = a p + b mod 26
a Z 26
gcd ( a , 26 ) = 1
a = 5
gcd ( 5 , 26 ) = 1
a 1 = 21
f ( p ) = 5 p + 3 mod 26
3 , 6 , 7 , 23 , 8 , 10 , 3
f 1 ( p ) = 21 p 21 3 mod 26 = 21 p + 15 mod 26
p 1
p 2
p = ( p 1 p 2 )
A
2 × 2
Z 26
f ( p ) = A p + b
b
Z 26
f 1 ( p ) = A 1 p A 1 b
7 , 4 , 11 , 15
A = ( 3 5 1 2 )
A 1 = ( 2 21 25 3 )
b = ( 2 , 2 ) t
f
f 1
p
q
n = p q
ϕ ( n ) = m = ( p 1 ) ( q 1 )
ϕ
ϕ
E
m
E
gcd ( E , m ) = 1
D
D E 1 ( mod m )
n
E
E
n
A = 00 , B = 02 , , Z = 25
n
x
y = x E mod n
y
x
x = y D mod n
D
25
p = 23
q = 29
n = p q = 667
ϕ ( n ) = m = ( p 1 ) ( q 1 ) = 616
E = 487
gcd ( 616 , 487 ) = 1
25 487 mod 667 = 169
191 E = 1 + 151 m
( n , D ) = ( 667 , 191 )
169 191 mod 667 = 25
D E 1 ( mod m )
k
D E = k m + 1 = k ϕ ( n ) + 1
gcd ( x , n ) = 1
y D = ( x E ) D = x D E = x k m + 1 = ( x ϕ ( n ) ) k x = ( 1 ) k x = x mod n
x
y D mod n
gcd ( x , n )   1
n = p q
x < n
x
p
q
r
r < q
x = r p
gcd ( x , q ) = 1
m = ϕ ( n ) = ( p 1 ) ( q 1 ) = ϕ ( p ) ϕ ( q )
q
x k m = x k ϕ ( p ) ϕ ( q ) = ( x ϕ ( q ) ) k ϕ ( p ) = ( 1 ) k ϕ ( p ) = 1 mod q
t
x k m = 1 + t q
y D = x k m + 1 = x k m x = ( 1 + t q ) x = x + t q ( r p ) = x + t r n = x mod n
D
n
E
n
D
667 = 23 29
D
( n , E )
( n , D )
( n , E )
( n , D )
x
x
x = x D mod n
x
x
x
x
y = x E mod n
ϕ ( 893 456 123 )
7 324 ( mod 895 )
26 ! 1
2 × 2
A
Z 26
gcd ( det ( A ) , 26 ) = 1
A = ( 3 4 2 3 )
f ( p ) = A p + b
b = ( 2 , 5 ) t
x
x
2
x = 142528
14
25
28
n = 3551 , E = 629 , x = 31
n = 2257 , E = 47 , x = 23
n = 120979 , E = 13251 , x = 142371
n = 45629 , E = 781 , x = 231561
2791
112135 25032 442
D
y
n = 3551 , D = 1997 , y = 2791
n = 5893 , D = 81 , y = 34
n = 120979 , D = 27331 , y = 112135
n = 79403 , D = 671 , y = 129381
31
14
( n , E )
D
( n , E ) = ( 451 , 231 )
( n , E ) = ( 3053 , 1921 )
( n , E ) = ( 37986733 , 12371 )
( n , E ) = ( 16394854313 , 34578451 )
n = 11 41
n = 8779 4327
n
n
n
E
X
X E X ( mod n )
10
15
( n , E )
D
n
n
n
d = 2 , 3 , , n
n
d
n
n
n = a b
n
n = x 2 y 2 = ( x y ) ( x + y )
x
y
n = x 2 y 2
n
p
gcd ( a , p ) = 1
a p 1 1 ( mod p )
15
2 15 1 2 14 4 ( mod 15 )
17
2 17 1 2 16 1 ( mod 17 )
n
2 n 1 1 ( mod n )
342
811
561
771
631
n
b
gcd ( b , n ) = 1
b n 1 1 ( mod n )
n
b
341
2
3
2000
2000
2000
561 = 3 11 17
25 × 10 9
n
21
ϕ
p q
( p 1 ) ( q 1 )
m
m
m
m
m
m
n
E
D
D
D
m
128 4 = 268 , 435 , 456
n
n
10 12
E
n
E
D
2 10 12
E
n
E
D
1
n
m
n
n
n
n
n
( x 1 , x 2 , , x n )
3 n
( x 1 , x 2 , , x n ) ( x 1 , x 2 , , x n , x 1 , x 2 , , x n , x 1 , x 2 , , x n )
i
i
( 0110 )
( 0110 0110 0110 )
( 0110 1110 0110 )
( 0110 )
n
n
2 n
n
m
m
3 n
8
2 8 = 256
8
2 7 = 128
A = 65 10 = 01000001 2 , B = 66 10 = 01000010 2 , C = 67 10 = 01000011 2
128
00000000 2 = 0 10 , 01111111 2 = 127 10
0
1
1
A = 01000001 2 , B = 01000010 2 , C = 11000011 2
( 0100 0101 )
1
m
8
16
32
0
1
1
1
1
0
1
( 1001 1000 )
0
1
0
( 000 )
1
( 111 )
( 101 )
1
0
( 111 )
000
001
010
011
100
101
110
111
000
0
1
1
2
1
2
2
3
111
3
2
2
1
2
1
1
0
( 000 )
( 111 )
3
p
q
p q
n
0
1
p
q = 1 p
1
1
p
0
q
n
p n
p = 0.999
( 0.999 ) 10 , 000 0.00005
n
( x 1 , , x n )
p
k
( n k ) q k p n k
k
k
q
n k
p
n
q k p n k
k
( n k ) = n ! k ! ( n k ) !
n
k
q k p n k
( n k ) q k p n k
p = 0.995
500
p n = ( 0.995 ) 500 0.082
( n 1 ) q p n 1 = 500 ( 0.005 ) ( 0.995 ) 499 0.204
( n 2 ) q 2 p n 2 = 500 499 2 ( 0.005 ) 2 ( 0.995 ) 498 0.257
1 0.082 0.204 0.257 = 0.457
( n , m )
m
n
( n , m )
E : Z 2 m Z 2 n
D : Z 2 n Z 2 m
E
E
D
( 8 , 7 )
E ( x 7 , x 6 , , x 1 ) = ( x 8 , x 7 , , x 1 )
x 8 = x 7 + x 6 + + x 1
Z 2
x = ( x 1 , , x n )
y = ( y 1 , , y n )
n
d ( x , y )
x
y
x
y
d min
d ( x , y )
x
y
w ( x )
x
1
x
w ( x ) = d ( x , 0 )
0 = ( 00 0 )
x
y
x
x = ( 10101 )
y = ( 11010 )
z = ( 00011 )
C
d ( x , y ) = 4 , d ( x , z ) = 3 , d ( y , z ) = 3
w ( x ) = 3 , w ( y ) = 3 , w ( z ) = 2
x
y
z
n
w ( x ) = d ( x , 0 )
d ( x , y ) 0
d ( x , y ) = 0
x = y
d ( x , y ) = d ( y , x )
d ( x , y ) d ( x , z ) + d ( z , y )
x = ( 1101 )
y = ( 1100 )
( 1101 )
( 1100 )
( 1100 )
d ( x , y ) = 1
x = ( 1100 )
y = ( 1010 )
d ( x , y ) = 2
x
y
2
0000
0011
0101
0110
1001
1010
1100
1111
0000
0
2
2
2
2
2
2
4
0011
2
0
2
2
2
2
4
2
0101
2
2
0
2
2
4
2
2
0110
2
2
2
0
4
2
2
2
1001
2
2
2
4
0
2
2
2
1010
2
2
4
2
2
0
2
2
1100
2
4
2
2
2
2
0
2
1111
4
2
2
2
2
2
2
0
x
y
d ( x , y ) = 1
x
y
x
y
y
d ( x , y ) = 2
x
y
d min = 2
d ( x , y ) = 2
y
z
d ( x , z ) = d ( y , z ) = 1
x
y
d min 3
x
y
d ( x , y ) = 1
d ( z , y ) 2
z   x
3
C
d min = 2 n + 1
C
n
2 n
C
x
y
n
d ( x , y ) n
z
x
2 n + 1 d ( x , z ) d ( x , y ) + d ( y , z ) n + d ( y , z )
d ( y , z ) n + 1
y
x
x
y
2 n
1 d ( x , y ) 2 n
2 n + 1
y
1
2 n
c 1 = ( 00000 )
c 2 = ( 00111 )
c 3 = ( 11100 )
c 4 = ( 11011 )
00000
00111
11100
11011
00000
0
3
3
4
00111
3
0
4
3
11100
3
4
0
3
11011
4
3
3
0
( 32 , 6 )
Z 2 n
n
n
0
( 11000101 ) + ( 11000101 ) = ( 00000000 )
( 0000000 ) ( 0001111 ) ( 0010101 ) ( 0011010 ) ( 0100110 ) ( 0101001 ) ( 0110011 ) ( 0111100 ) ( 1000011 ) ( 1001100 ) ( 1010110 ) ( 1011001 ) ( 1100101 ) ( 1101010 ) ( 1110000 ) ( 1111111 )
Z 2 7
d min = 3
3
x
y
n
w ( x + y ) = d ( x , y )
x
y
n
x
y
x
y
x
y
1
1 + 1 = 0 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1
d min
C
d min
C
d min = min { w ( x ) : x   0 }
d min = min { d ( x , y ) : x   y } = min { d ( x , y ) : x + y   0 } = min { w ( x + y ) : x + y   0 } = min { w ( z ) : z   0 }
3
n
x y = x 1 y 1 + + x n y n
x = ( x 1 , x 2 , , x n ) t
y = ( y 1 , y 2 , , y n ) t
n
x = ( 011001 ) t
y = ( 110101 ) t
x y = 0
x y = x t y = ( x 1 x 2 x n ) ( y 1 y 2 y n ) = x 1 y 1 + x 2 y 2 + + x n y n
3
3
1
n
x = ( x 1 , x 2 , , x n ) t
1
x 1 + x 2 + + x n = 0
4
x = ( x 1 , x 2 , x 3 , x 4 ) t
1
x 1 + x 2 + x 3 + x 4 = 0
x 1 = x t 1 = ( x 1 x 2 x 3 x 4 ) ( 1 1 1 1 ) = 0
M m × n ( Z 2 )
m × n
Z 2
Z 2
H M m × n ( Z 2 )
n
x
H x = 0
H
Null ( H )
m × n
Z 2
H
H = ( 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 )
5
x = ( x 1 , x 2 , x 3 , x 4 , x 5 ) t
H
H x = 0
x 2 + x 4 = 0 x 1 + x 2 + x 3 + x 4 = 0 x 3 + x 4 + x 5 = 0
5
( 00000 ) ( 11110 ) ( 10101 ) ( 01011 )
H
M m × n ( Z 2 )
H
Z 2 n
x , y Null ( H )
H
M m × n ( Z 2 )
H x = 0
H y = 0
H ( x + y ) = H x + H y = 0 + 0 = 0
x + y
H
H M m × n ( Z 2 )
C
H = ( 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 )
6
x = ( 010011 ) t
x
H x = ( 0 1 1 )
H
H
H
m × n
Z 2
n > m
m
m × m
I m
H = ( A I m )
A
m × ( n m )
( a 11 a 12 a 1 , n m a 21 a 22 a 2 , n m a m 1 a m 2 a m , n m )
I m
m × m
( 1 0 0 0 1 0 0 0 1 )
n × ( n m )
G = ( I n m A )
x
G x = y
H y = 0
x
G
y
( 000 ) , ( 001 ) , ( 010 ) , , ( 111 )
A = ( 0 1 1 1 1 0 1 0 1 )
G = ( 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 )
H = ( 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 )
H
6
1
1
x = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 )
0 = H x = ( x 2 + x 3 + x 4 x 1 + x 2 + x 5 x 1 + x 3 + x 6 )
x 2 + x 3 + x 4 = 0 x 1 + x 2 + x 5 = 0 x 1 + x 3 + x 6 = 0
x 4
x 2
x 3
x 5
x 1
x 2
x 6
x 1
x 3
x 4
x 5
x 6
x 1
x 2
x 3
x 4
x 5
x 6
H
( 000000 ) ( 001101 ) ( 010110 ) ( 011011 ) ( 100011 ) ( 101110 ) ( 110101 ) ( 111000 ) .
G
x
G x
000
000000
001
001101
010
010110
011
011011
100
100011
101
101110
110
110101
111
111000
H M m × n ( Z 2 )
Null ( H )
x Z 2 n
n m
m
H x = 0
m
n m
H
( n , n m )
n m
x
m
G
n × k
C = { y : G x = y for x Z 2 k }
( n , k )
C
G x 1 = y 1
G x 2 = y 2
y 1 + y 2
C
G ( x 1 + x 2 ) = G x 1 + G x 2 = y 1 + y 2
G x = G y
x = y
G x = G y
G x G y = G ( x y ) = 0
k
G ( x y )
x 1 y 1 , , x k y k
I k
G
G ( x y ) = 0
x = y
H = ( A I m )
m × n
G = ( I n m A )
n × ( n m )
H G = 0
C = H G
i j
C
c i j = k = 1 n h i k g k j = k = 1 n m h i k g k j + k = n m + 1 n h i k g k j = k = 1 n m a i k δ k j + k = n m + 1 n δ i ( m n ) , k a k j = a i j + a i j = 0
δ i j = { 1 i = j 0 i   j
H = ( A I m )
m × n
G = ( I n m A )
n × ( n m )
H
C
G
y
C
H y = 0
C
H
y C
G x = y
x Z 2 m
H y = H G x = 0
y = ( y 1 , , y n ) t
H
x
Z 2 n m
G x t = y
H y = 0
a 11 y 1 + a 12 y 2 + + a 1 , n m y n m + y n m + 1 = 0 a 21 y 1 + a 22 y 2 + + a 2 , n m y n m + y n m + 1 = 0 a m 1 y 1 + a m 2 y 2 + + a m , n m y n m + y n m + 1 = 0
y n m + 1 , , y n
y 1 , , y n m
y n m + 1 = a 11 y 1 + a 12 y 2 + + a 1 , n m y n m y n m + 1 = a 21 y 1 + a 22 y 2 + + a 2 , n m y n m y n m + 1 = a m 1 y 1 + a m 2 y 2 + + a m , n m y n m
x i = y i
i = 1 , , n m
H
e 1 = ( 100 00 ) t e 2 = ( 010 00 ) t e n = ( 000 01 ) t
n
Z 2 n
1
m × n
H
H e i
i
H
( 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 ) ( 0 1 0 0 0 ) = ( 1 0 1 )
e i
n
1
i
0
H M m × n ( Z 2 )
H e i
i
H
H
m × n
H
H
Null ( H )
2
2
e i
i = 1 , , n
H e i
i
H
e i
H
i
H
H e i   0
H 1 = ( 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 )
H 2 = ( 1 1 1 0 0 1 0 0 0 0 1 1 0 0 1 )
H 1
H 2
H
H
2
3
H = ( 1 1 1 0 1 0 0 1 1 1 0 0 )
H
Null ( H )
4
2
( 1100 )
( 1010 )
( 1001 )
( 0110 )
( 0101 )
( 0011 )
Null ( H )
H
H
H
H
H
H
H
n
e i + e j
1
i
j
w ( e i + e j ) = 2
i   j
0 = H ( e i + e j ) = H e i + H e j
i
j
H
H
2 3 = 8
( 0 0 0 ) , ( 1 0 0 ) , ( 0 1 0 ) , ( 0 0 1 )
3
H
m × n
n m
m
2 m
0 , e 1 , , e m
2 m ( 1 + m )
n
n
H = ( 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 )
5
x = ( 11011 ) t
y = ( 01011 ) t
H x = ( 0 0 0 ) and H y = ( 1 0 1 )
x
y
x
y
H y
H
y
0
1
x
H
m × n
x Z 2 n
x
H x
m × n
H
x
n
x
x = c + e
c
e
H x
x
e
H x = H ( c + e ) = H c + H e = 0 + H e = H e
H e
i
H
H M m × n ( Z 2 )
H
r
n
r
0
r
H
i
i
H = ( 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 )
6
x = ( 111110 ) t
y = ( 111111 ) t
z = ( 010111 ) t
H x = ( 1 1 1 ) , H y = ( 1 1 0 ) , H z = ( 1 0 0 )
x
z
x
z
( 110110 )
( 010011 )
y
H
y
C
Z 2 n
C
Z 2 n
C
( n , m )
C
Z 2 n
x + C
x Z 2 n
2 n m
C
Z 2 n
C
( 5 , 3 )
H = ( 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 )
( 00000 ) ( 01101 ) ( 10011 ) ( 11110 )
2 5 2 = 2 3
C
Z 2 5
2 2 = 4
C
C
( 00000 ) ( 01101 ) ( 10011 ) ( 11110 )
( 10000 ) + C
( 10000 ) ( 11101 ) ( 00011 ) ( 01110 )
( 01000 ) + C
( 01000 ) ( 00101 ) ( 11011 ) ( 10110 )
( 00100 ) + C
( 00100 ) ( 01001 ) ( 10111 ) ( 11010 )
( 00010 ) + C
( 00010 ) ( 01111 ) ( 10001 ) ( 11100 )
( 00001 ) + C
( 00001 ) ( 01100 ) ( 10010 ) ( 11111 )
( 10100 ) + C
( 00111 ) ( 01010 ) ( 10100 ) ( 11001 )
( 00110 ) + C
( 00110 ) ( 01011 ) ( 10101 ) ( 11000 )
x
r
n
e
r = e + x
x = e + r
r
e + C
e
e
n
r + e
x
r = ( 01111 )
r
( 00010 ) + C
( 01101 ) = ( 01111 ) + ( 00010 )
( 000 )
( 00000 )
( 001 )
( 00001 )
( 010 )
( 00010 )
( 011 )
( 10000 )
( 100 )
( 00100 )
( 101 )
( 01000 )
( 110 )
( 00110 )
( 111 )
( 10100 )
C
( n , k )
H
x
y
Z 2 n
x
y
C
H x = H y
n
n
x
y
C
x y C
H ( x y ) = 0
H x = H y
C
x = ( 01111 )
H x = ( 0 1 0 )
( 00010 )
( n , k )
2 n k
C
( 32 , 24 )
2 24
2 32 24 = 2 8 = 256
d = 6
56 10
56 10
C
H = [ 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 ]
x = 11100
x
C
H
x
x
0
1
2
3
4
5
6
7
8
000
001
010
011
101
110
111
000
001
4
Z 2 4
( 0110 ) ( 1001 ) ( 1010 ) ( 1100 )
( 0000 ) C
n
( 011010 ) , ( 011100 )
( 11110101 ) , ( 01010100 )
( 00110 ) , ( 01111 )
( 1001 ) , ( 0111 )
2
2
n
( 011010 )
( 11110101 )
( 01111 )
( 1011 )
3
4
C
7
C
( 011010 ) ( 011100 ) ( 110111 ) ( 110000 )
( 011100 ) ( 011011 ) ( 111011 ) ( 100011 ) ( 000000 ) ( 010101 ) ( 110100 ) ( 110011 )
( 000000 ) ( 011100 ) ( 110101 ) ( 110001 )
( 0110110 ) ( 0111100 ) ( 1110000 ) ( 1111111 ) ( 1001001 ) ( 1000011 ) ( 0001111 ) ( 0000000 )
d min = 2
d min = 1
( n , k )
( 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 )
( 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 )
( 1 0 0 1 1 0 1 0 1 1 )
( 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 )
( 00000 ) , ( 00101 ) , ( 10011 ) , ( 10110 )
G = ( 0 1 0 0 1 0 0 1 1 1 )
( 000000 ) , ( 010111 ) , ( 101101 ) , ( 111010 )
G = ( 1 0 0 1 1 0 1 1 0 1 1 1 )
( 5 , 2 )
C
H = ( 0 1 0 0 1 1 0 1 0 1 0 0 1 1 1 )
01111 10101 01110 00011
1000
p
p = 0.01
p = 0.0001
( 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 )
( 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 1 )
( 1 1 1 0 1 0 0 1 )
( 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 )
G = ( 1 1 0 0 1 )
G = ( 1 0 0 1 1 1 1 0 )
H = ( 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 )
C
Z 2 3
( 000 )
( 111 )
C
Z 2 3
C
( 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 )
( 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 )
( 1 0 0 1 1 0 1 0 1 1 )
( 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 )
C
( 10000 ) + C
( 01000 ) + C
( 00100 ) + C
( 00010 ) + C
( 11000 ) + C
( 01100 ) + C
( 01010 ) + C
C
x
y
z
n
w ( x ) = d ( x , 0 )
d ( x , y ) = d ( x + z , y + z )
d ( x , y ) = w ( x y )
X
d : X × X R
d ( x , y ) 0
x , y X
d ( x , y ) = 0
x = y
d ( x , y ) = d ( y , x )
d ( x , y ) d ( x , z ) + d ( z , y )
Z 2 n
C
i
C
C
x C
y x + y
C
20
e i
n
1
i
0
H M m × n ( Z 2 )
H e i
i
H
C
( n , k )
C
C = { x Z 2 n : x y = 0 for all y C }
C
C
( 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 )
C
( n , n k )
C
C
H
m × n
Z 2
i
i
m
H = ( 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 0 )
i
i
( 101011 )
H
( 101101 )
( 001001 )
( 0010000101 )
( 0000101100 )
( m , n )
k
0
( 16 , 12 )
( 7 , 4 )
7
7
4
2 4 = 16
7
d = 3
7
3
( n , k )
r
r
d
( d 1 ) / 2
r
n
1 + ( n 1 ) + ( n 2 ) + + ( n r )
d
k
2 k ( ( n 0 ) + ( n 1 ) + ( n 2 ) + + ( n d 1 2 ) ) = 2 n
d
32
Z 4
T
i
( G , )
( H , )
ϕ : G H
ϕ ( a b ) = ϕ ( a ) ϕ ( b )
a
b
G
G
H
G H
G
H
ϕ
Z 4 i
ϕ : Z 4 i
ϕ ( n ) = i n
ϕ
ϕ
ϕ ( 0 ) = 1 ϕ ( 1 ) = i ϕ ( 2 ) = 1 ϕ ( 3 ) = i
ϕ ( m + n ) = i m + n = i m i n = ϕ ( m ) ϕ ( n )
ϕ
( R , + )
( R + , )
ϕ ( x + y ) = e x + y = e x e y = ϕ ( x ) ϕ ( y )
ϕ
Q
2 n
ϕ : Z Q
ϕ ( n ) = 2 n
ϕ ( m + n ) = 2 m + n = 2 m 2 n = ϕ ( m ) ϕ ( n )
ϕ
{ 2 n : n Z }
Q
m   n
ϕ ( m )   ϕ ( n )
m > n
ϕ ( m ) = ϕ ( n )
2 m = 2 n
2 m n = 1
m n > 0
Z 8
Z 12
U ( 8 ) U ( 12 )
U ( 8 ) = { 1 , 3 , 5 , 7 } U ( 12 ) = { 1 , 5 , 7 , 11 }
ϕ : U ( 8 ) U ( 12 )
1 1 3 5 5 7 7 11
ϕ
ψ
ψ ( 1 ) = 1
ψ ( 3 ) = 11
ψ ( 5 ) = 5
ψ ( 7 ) = 7
Z 2 × Z 2
S 3
Z 6
Z 6
S 3
ϕ : Z 6 S 3
a , b S 3
a b   b a
ϕ
m
n
Z 6
ϕ ( m ) = a and ϕ ( n ) = b
a b = ϕ ( m ) ϕ ( n ) = ϕ ( m + n ) = ϕ ( n + m ) = ϕ ( n ) ϕ ( m ) = b a
a
b
ϕ : G H
ϕ 1 : H G
| G | = | H |
G
H
G
H
G
n
H
n
ϕ
h 1
h 2
H
ϕ
g 1 , g 2 G
ϕ ( g 1 ) = h 1
ϕ ( g 2 ) = h 2
h 1 h 2 = ϕ ( g 1 ) ϕ ( g 2 ) = ϕ ( g 1 g 2 ) = ϕ ( g 2 g 1 ) = ϕ ( g 2 ) ϕ ( g 1 ) = h 2 h 1
Z
G
a
G
ϕ : Z G
ϕ : n a n
ϕ ( m + n ) = a m + n = a m a n = ϕ ( m ) ϕ ( n )
ϕ
m
n
Z
m   n
m > n
a m   a n
a m = a n
a m n = e
m n > 0
a
G
a n
n
ϕ ( n ) = a n
G
n
G
Z n
G
n
a
ϕ : Z n G
ϕ : k a k
0 k < n
ϕ
G
p
p
G
Z p
G
G
Z 3
Z 3
+ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1
Z 3
G = { ( 0 ) , ( 0 1 2 ) , ( 0 2 1 ) }
0 ( 0 1 2 0 1 2 ) = ( 0 ) 1 ( 0 1 2 1 2 0 ) = ( 0 1 2 ) 2 ( 0 1 2 2 0 1 ) = ( 0 2 1 )
G
G ¯
G
g G
λ g : G G
λ g ( a ) = g a
λ g
G
λ g
λ g ( a ) = λ g ( b )
g a = λ g ( a ) = λ g ( b ) = g b
a = b
λ g
a G
b
λ g ( b ) = a
b = g 1 a
G ¯
G ¯ = { λ g : g G }
G ¯
G
G ¯
( λ g λ h ) ( a ) = λ g ( h a ) = g h a = λ g h ( a )
λ e ( a ) = e a = a
( λ g 1 λ g ) ( a ) = λ g 1 ( g a ) = g 1 g a = a = λ e ( a )
G
G ¯
ϕ : g λ g
ϕ ( g h ) = λ g h = λ g λ h = ϕ ( g ) ϕ ( h )
ϕ ( g ) ( a ) = ϕ ( h ) ( a )
g a = λ g a = λ h a = h a
g = h
ϕ
ϕ ( g ) = λ g
λ g G ¯
g λ g
G
G
H
G
H
G × H
G
G
( G , )
( H , )
G
H
( g , h ) G × H
g G
h H
G × H
( g 1 , h 1 ) ( g 2 , h 2 ) = ( g 1 g 2 , h 1 h 2 ) ;
G
H
( g 1 , h 1 ) ( g 2 , h 2 ) = ( g 1 g 2 , h 1 h 2 )
G
H
G × H
( g 1 , h 1 ) ( g 2 , h 2 ) = ( g 1 g 2 , h 1 h 2 )
g 1 , g 2 G
h 1 , h 2 H
e G
e H
G
H
( e G , e H )
G × H
( g , h ) G × H
( g 1 , h 1 )
G
H
R
R
R × R = R 2
( a , b ) + ( c , d ) = ( a + c , b + d )
( 0 , 0 )
( a , b )
( a , b )
Z 2 × Z 2 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) }
Z 2 × Z 2
Z 4
( a , b )
Z 2 × Z 2
2
( a , b ) + ( a , b ) = ( 0 , 0 )
Z 4
G × H
G
H
i = 1 n G i = G 1 × G 2 × × G n
G 1 , G 2 , , G n
G = G 1 = G 2 = = G n
G n
G 1 × G 2 × × G n
Z 2 n
n
n
( 01011101 ) + ( 01001011 ) = ( 00010110 )
( g , h ) G × H
g
h
r
s
( g , h )
G × H
r
s
m
r
s
n = | ( g , h ) |
( g , h ) m = ( g m , h m ) = ( e G , e H ) ( g n , h n ) = ( g , h ) n = ( e G , e H )
n
m
n m
r
s
n
n
r
s
m
r
s
m n
m
n
( g 1 , , g n ) G i
g i
r i
G i
( g 1 , , g n )
G i
r 1 , , r n
( 8 , 56 ) Z 12 × Z 60
gcd ( 8 , 12 ) = 4
8
12 / 4 = 3
Z 12
56
Z 60
15
3
15
15
( 8 , 56 )
15
Z 12 × Z 60
Z 2 × Z 3
( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 0 ) , ( 1 , 1 ) , ( 1 , 2 )
Z 2 × Z 2
Z 4
Z 2 × Z 3 Z 6
Z 2 × Z 3
( 1 , 1 )
Z 2 × Z 3
Z m × Z n
Z m n
gcd ( m , n ) = 1
Z m × Z n Z m n
gcd ( m , n ) = 1
gcd ( m , n ) = d > 1
Z m × Z n
m n / d
m
n
( a , b ) Z m × Z n
( a , b ) + ( a , b ) + + ( a , b ) m n / d times = ( 0 , 0 )
( a , b )
Z m × Z n
lcm ( m , n ) = m n
gcd ( m , n ) = 1
n 1 , , n k
i = 1 k Z n i Z n 1 n k
gcd ( n i , n j ) = 1
i   j
m = p 1 e 1 p k e k
p i
Z m Z p 1 e 1 × × Z p k e k
p i e i
p j e j
i   j
Z p 1 e 1 × × Z p k e k
p 1 , , p k
G
H
K
G = H K = { h k : h H , k K }
H K = { e }
h k = k h
k K
h H
G
H
K
U ( 8 )
H = { 1 , 3 } and K = { 1 , 5 }
D 6
H = { i d , r 3 } and K = { i d , r 2 , r 4 , s , r 2 s , r 4 s }
K S 3
D 6 Z 2 × S 3
S 3
H
K
H
3
H
{ ( 1 ) , ( 123 ) , ( 132 ) }
K
2
K
h k = k h
h H
k K
G
H
K
G
H × K
G
g G
g = h k
h H
k K
ϕ : G H × K
ϕ ( g ) = ( h , k )
ϕ
h
k
g
g = h k = h k
h 1 h = k ( k ) 1
H
K
h = h
k = k
ϕ
ϕ
g 1 = h 1 k 1
g 2 = h 2 k 2
ϕ ( g 1 g 2 ) = ϕ ( h 1 k 1 h 2 k 2 ) = ϕ ( h 1 h 2 k 1 k 2 ) = ( h 1 h 2 , k 1 k 2 ) = ( h 1 , k 1 ) ( h 2 , k 2 ) = ϕ ( g 1 ) ϕ ( g 2 )
ϕ
Z 6
{ 0 , 2 , 4 } × { 0 , 3 }
G
H 1 , H 2 , , H n
G
G = H 1 H 2 H n = { h 1 h 2 h n : h i H i }
H i j   i H j = { e }
h i h j = h j h i
h i H i
h j H j
G
H i
i = 1 , 2 , , n
G
i H i
( 1 , 2 )
Z 4 × Z 8
Z 15
Z n Z
n   0
Z
C
G L 2 ( R )
( a b b a )
ϕ : C G L 2 ( R )
ϕ ( a + b i ) = ( a b b a )
U ( 8 ) Z 4
U ( 8 )
( 1 0 0 1 ) , ( 1 0 0 1 ) , ( 1 0 0 1 ) , ( 1 0 0 1 )
U ( 5 )
U ( 10 )
U ( 12 )
n
Z n
Z n
n
k cis ( 2 k π / n )
n
Z n
Q
Z
Q
G = R { 1 }
G
a b = a + b + a b
G
( G , )
( 1 0 0 0 1 0 0 0 1 ) ( 1 0 0 0 0 1 0 1 0 ) ( 0 1 0 1 0 0 0 0 1 ) ( 0 0 1 1 0 0 0 1 0 ) ( 0 0 1 0 1 0 1 0 0 ) ( 0 1 0 0 0 1 1 0 0 )
G
6
8
S 4
D 12
ω = cis ( 2 π / n )
n
A = ( ω 0 0 ω 1 ) and B = ( 0 1 1 0 )
D n
( ± 1 k 0 1 )
D n
Z n
Z 4 × Z 2
( 3 , 4 )
Z 4 × Z 6
( 6 , 15 , 4 )
Z 30 × Z 45 × Z 24
( 5 , 10 , 15 )
Z 25 × Z 25 × Z 25
( 8 , 8 , 8 )
Z 10 × Z 24 × Z 80
12
5
D 4
Q
2 m 3 n
m , n Z
Z × Z
S 3 × Z 2
D 6
D 2 n
3
3
6
G
20
G
H
K
4
5
h k = k h
h H
k K
G
H
K
G
H
K
G × K H × K
G H
51
52
ϕ : G H
ϕ ( x ) = e H
x = e G
e G
e H
G
H
G H
G
H
a
G
ϕ : G H
ϕ ( a )
H
G
p
p
Z p
S n
A n + 2
D n
S n
ϕ : G 1 G 2
ψ : G 2 G 3
ϕ 1
ψ ϕ
U ( 5 ) Z 4
U ( p )
p
S 3
G
ϕ ( a + b i ) = a b i
C
C
a + i b a i b
C
A B 1 A B
S L 2 ( R )
B
G L 2 ( R )
G
Aut ( G )
G
Aut ( G )
S G
G
Aut ( Z 6 )
Z 6
Z 6
Aut ( Z )
G
H
Aut ( G ) Aut ( H )
G
g G
i g : G G
i g ( x ) = g x g 1
i g
G
Inn ( G )
i g ( x ) = g x g 1
G
Inn ( G )
Aut ( G )
Q 8
Inn ( G ) = Aut ( G )
G
g G
λ g : G G
ρ g : G G
λ g ( x ) = g x
ρ g ( x ) = x g 1
i g = ρ g λ g
G
g ρ g
G
G
H
K
ϕ : G H × K
ϕ ( g ) = ( h , k )
g = h k
h H
k K
ϕ
g 1 = h 1 k 1
g 2 = h 2 k 2
ϕ ( g 1 ) = ϕ ( g 2 )
G
H
G
n
H
n
G G ¯
H H ¯
G × H G ¯ × H ¯
G × H
H × G
n 1 , , n k
i = 1 k Z n i Z n 1 n k
gcd ( n i , n j ) = 1
i   j
A × B
A
B
G
H 1 , H 2 , , H n
G
i H i
H 1
H 2
G 1
G 2
H 1 × H 2
G 1 × G 2
m , n Z
m , n = d
d = gcd ( m , n )
m , n Z
m n = l
l = lcm ( m , n )
2 p
2 p
p
G
2 p
p
a G
a
1
2
p
2 p
G
2 p
G
Z 2 p
G
G
2 p
G
p
G
p
G
2 p
G
2
P
G
p
y G
2
y P = P y
G
2 p
P = z
p
z
y
2
y z = z k y
2 k < p
G
2 p
G
G
2 p
P = z
p
z
y
2
G
{ z i y j 0 i < p , 0 j < 2 }
G
2 p
P = z
p
z
y
2
( z i y j ) ( z r y s )
z m y n
m , n
2 p
16
Z n
Z p
16
1 , 2 , 3 , 5 , 7 , 11
13
4
Z 4
Z 2 × Z 2
4
4
4
6
Z 3 × Z 2
Z 6
2
3
D 3
S 3
3
2 p
p
Z 6
D 3
6
6
10
14
n
n
n
n
12
6
2
6
6
2
6
6
6
2
6
2
Z 6 × Z 2
Z 3 × Z 2 × Z 2
3
2
2
Q
Q
± 1 , ± I , ± J , ± K
S 8
g Q
T g
T g ( x ) = x g
T g
Q
{ 1 , 2 , , 8 }
Z 2 × Z 4
8
Z 2 × Z 4
Z 2 × Z 4
U ( 24 )
U ( 24 )
D 10
1
10
180
R
2
72
10
S
10
R
S
G
G
H
G
g H = H g
g G
H
G
g H = H g
g G
G
G
H
G
g h = h g
g G
h H
g H = H g
H
S 3
( 1 )
( 12 )
( 123 ) H = { ( 123 ) , ( 13 ) } and H ( 123 ) = { ( 123 ) , ( 23 ) }
H
S 3
N
( 1 )
( 123 )
( 132 )
N
N = { ( 1 ) , ( 123 ) , ( 132 ) } ( 12 ) N = N ( 12 ) = { ( 12 ) , ( 13 ) , ( 23 ) }
G
N
G
N
G
g G
g N g 1 N
g G
g N g 1 = N
N
G
g N = N g
g G
g G
n N
n
N
g n = n g
g n g 1 = n N
g N g 1 N
g G
g N g 1 N
N g N g 1
n N
g 1 n g = g 1 n ( g 1 ) 1 N
g 1 n g = n
n N
n = g n g 1
g N g 1
g N g 1 = N
g G
n N
n N
g n g 1 = n
g n = n g
g N N g
N g g N
N
G
N
G
G / N
( a N ) ( b N ) = a b N
G
N
G
N
G / N
N
G
N
G
G / N
[ G : N ]
G / N
( a N ) ( b N ) = a b N
a N = b N
c N = d N
( a N ) ( c N ) = a c N = b d N = ( b N ) ( d N )
a = b n 1
c = d n 2
n 1
n 2
N
a c N = b n 1 d n 2 N = b n 1 d N = b n 1 N d = b N d = b d N
e N = N
g 1 N
g N
G / N
N
G
S 3
N = { ( 1 ) , ( 123 ) , ( 132 ) }
N
S 3
N
( 12 ) N
S 3 / N
N ( 12 ) N N N ( 12 ) N ( 12 ) N ( 12 ) N N
Z 2
S 3 / N
S 3
N = A 3
( 12 ) N = { ( 12 ) , ( 13 ) , ( 23 ) }
G / N
3 Z
Z
3 Z
Z
0 + 3 Z = { , 3 , 0 , 3 , 6 , } 1 + 3 Z = { , 2 , 1 , 4 , 7 , } 2 + 3 Z = { , 1 , 2 , 5 , 8 , }
Z / 3 Z
+ 0 + 3 Z 1 + 3 Z 2 + 3 Z 0 + 3 Z 0 + 3 Z 1 + 3 Z 2 + 3 Z 1 + 3 Z 1 + 3 Z 2 + 3 Z 0 + 3 Z 2 + 3 Z 2 + 3 Z 0 + 3 Z 1 + 3 Z
n Z
Z
Z / n Z
n Z 1 + n Z 2 + n Z ( n 1 ) + n Z
k + n Z
l + n Z
k + l + n Z
D n
r
s
r n = i d s 2 = i d s r s = r 1
r
R n
D n
s r s 1 = s r s = r 1 R n
D n
D n / R n
Z 2
Z p
p
A n
n 5
A n
3
n 3
3
A n
3
( a b ) = ( b a )
( a b ) ( a b ) = i d ( a b ) ( c d ) = ( a c b ) ( a c d ) ( a b ) ( a c ) = ( a c b )
N
A n
n 3
N
3
N = A n
A n
3
( i j k )
i
j
{ 1 , 2 , , n }
k
3
3
( i a j ) = ( i j a ) 2 ( i a b ) = ( i j b ) ( i j a ) 2 ( j a b ) = ( i j b ) 2 ( i j a ) ( a b c ) = ( i j a ) 2 ( i j c ) ( i j b ) 2 ( i j a )
N
A n
n 3
N
3
( i j a )
N
[ ( i j ) ( a k ) ] ( i j a ) 2 [ ( i j ) ( a k ) ] 1 = ( i j k )
N
N
3
( i j k )
1 k n
3
A n
N = A n
n 5
N
A n
3
σ
N
σ
σ
3
σ
σ = τ ( a 1 a 2 a r ) N
r > 3
σ
σ = τ ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 )
σ = τ ( a 1 a 2 a 3 )
τ
σ = τ ( a 1 a 2 ) ( a 3 a 4 )
τ
σ
3
N
σ
σ = τ ( a 1 a 2 a r )
( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1
N
N
σ 1 ( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1
N
σ 1 ( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1 = σ 1 ( a 1 a 2 a 3 ) σ ( a 1 a 3 a 2 ) = ( a 1 a 2 a r ) 1 τ 1 ( a 1 a 2 a 3 ) τ ( a 1 a 2 a r ) ( a 1 a 3 a 2 ) = ( a 1 a r a r 1 a 2 ) ( a 1 a 2 a 3 ) ( a 1 a 2 a r ) ( a 1 a 3 a 2 ) = ( a 1 a 3 a r )
N
3
N = A n
N
σ = τ ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 )
σ 1 ( a 1 a 2 a 4 ) σ ( a 1 a 2 a 4 ) 1 N
( a 1 a 2 a 4 ) σ ( a 1 a 2 a 4 ) 1 N
σ 1 ( a 1 a 2 a 4 ) σ ( a 1 a 2 a 4 ) 1 = [ τ ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 ) ] 1 ( a 1 a 2 a 4 ) τ ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 ) ( a 1 a 2 a 4 ) 1 = ( a 4 a 6 a 5 ) ( a 1 a 3 a 2 ) τ 1 ( a 1 a 2 a 4 ) τ ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 ) ( a 1 a 4 a 2 ) = ( a 4 a 6 a 5 ) ( a 1 a 3 a 2 ) ( a 1 a 2 a 4 ) ( a 1 a 2 a 3 ) ( a 4 a 5 a 6 ) ( a 1 a 4 a 2 ) = ( a 1 a 4 a 2 a 6 a 3 )
N
N
σ = τ ( a 1 a 2 a 3 )
τ
σ N
σ 2 N
σ 2 = τ ( a 1 a 2 a 3 ) τ ( a 1 a 2 a 3 ) = ( a 1 a 3 a 2 )
N
3
σ = τ ( a 1 a 2 ) ( a 3 a 4 )
τ
2
σ 1 ( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1
N
( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1
N
σ 1 ( a 1 a 2 a 3 ) σ ( a 1 a 2 a 3 ) 1 = τ 1 ( a 1 a 2 ) ( a 3 a 4 ) ( a 1 a 2 a 3 ) τ ( a 1 a 2 ) ( a 3 a 4 ) ( a 1 a 2 a 3 ) 1 = ( a 1 a 3 ) ( a 2 a 4 )
n 5
b { 1 , 2 , , n }
b   a 1 , a 2 , a 3 , a 4
μ = ( a 1 a 3 b )
μ 1 ( a 1 a 3 ) ( a 2 a 4 ) μ ( a 1 a 3 ) ( a 2 a 4 ) N
μ 1 ( a 1 a 3 ) ( a 2 a 4 ) μ ( a 1 a 3 ) ( a 2 a 4 ) = ( a 1 b a 3 ) ( a 1 a 3 ) ( a 2 a 4 ) ( a 1 a 3 b ) ( a 1 a 3 ) ( a 2 a 4 ) = ( a 1 a 3 b )
N
3
A n
n 5
N
A n
N
3
N = A n
A n
n 5
A 5
196,833 × 196,833
G
1 , 2 , 3
H
H = ( 1 2 )
H
G
H
G
8 Z
Z
Z / 8 Z
( 3 + 8 Z ) + ( 7 + 8 Z )
G
H
H
G
G
H
G
H
G / H
G = S 4
H = A 4
G = A 5
H = { ( 1 ) , ( 123 ) , ( 132 ) }
G = S 4
H = D 4
G = Q 8
H = { 1 , 1 , I , I }
G = Z
H = 5 Z
A 4 ( 12 ) A 4 A 4 A 4 ( 12 ) A 4 ( 12 ) A 4 ( 12 ) A 4 A 4
D 4
S 4
D 4
D 4
Q 8
Q 8
T
2 × 2
R
( a b 0 c )
a
b
c R
a c   0
U
( 1 x 0 1 )
x R
U
T
U
U
T
T / U
T
G L 2 ( R )
G
G / H
H
G
H
G / H
G
G
G / H
a G
G
a H
G / H
H
G / H
G
H
2
G
H
G
S n
n 3
G
H
k
H
G
g G
i g : G G
i g : x g x g 1
G
i g ( H )
g
G
C ( g ) = { x G : x g = g x }
C ( g )
G
g
G
C ( g )
G
g
G
y
G
x C ( g )
y x y 1
C ( g )
( y x y 1 ) g = g ( y x y 1 )
G
Z ( G ) = { x G : x g = g x for all g G }
S 3
G L 2 ( R )
G
G
G / Z ( G )
G
G
G = a b a 1 b 1
G
G
a b a 1 b 1
G
G
G
G
G
N
G
G / N
N
G
g G
h G
h = a b a 1 b 1
g h g 1 = g a b a 1 b 1 g 1 = ( g a g 1 ) ( g b g 1 ) ( g a 1 g 1 ) ( g b 1 g 1 ) = ( g a g 1 ) ( g b g 1 ) ( g a g 1 ) 1 ( g b g 1 ) 1
h = h 1 h n
h i = a i b i a i 1 b i 1
g h g 1
g h g 1 = g h 1 h n g 1 = ( g h 1 g 1 ) ( g h 2 g 1 ) ( g h n g 1 )
D 8
8
1
3
4
1
3
1
3
2
1
3
2
1
3
1
3
A 4
1
3
3
1
3
0
2
A 5
A 5
A 5
A 5
4
4
4
4
Z 2 × Z 2
4
A 5
8
4
4
8
4
8
3
2 n
n
D n
3 n 100
D 470448
D 470448
( G , )
( H , )
ϕ : G H
ϕ ( g 1 g 2 ) = ϕ ( g 1 ) ϕ ( g 2 )
g 1 , g 2 G
ϕ
H
ϕ
S n
Z 2
S n
Z 2
even odd even even odd odd odd even
G
g G
ϕ : Z G
ϕ ( n ) = g n
ϕ
ϕ ( m + n ) = g m + n = g m g n = ϕ ( m ) ϕ ( n )
Z
G
g
G = G L 2 ( R )
A = ( a b c d )
G
det ( A ) = a d b c   0
A
B
G
det ( A B ) = det ( A ) det ( B )
ϕ : G L 2 ( R ) R
A det ( A )
T
z
| z | = 1
ϕ
R
T
ϕ : θ cos θ + i sin θ
ϕ ( α + β ) = cos ( α + β ) + i sin ( α + β ) = ( cos α cos β sin α sin β ) + i ( sin α cos β + cos α sin β ) = ( cos α + i sin α ) ( cos β + i sin β ) = ϕ ( α ) ϕ ( β )
ϕ : G 1 G 2
e
G 1
ϕ ( e )
G 2
g G 1
ϕ ( g 1 ) = [ ϕ ( g ) ] 1
H 1
G 1
ϕ ( H 1 )
G 2
H 2
G 2
ϕ 1 ( H 2 ) = { g G 1 : ϕ ( g ) H 2 }
G 1
H 2
G 2
ϕ 1 ( H 2 )
G 1
e
e
G 1
G 2
e ϕ ( e ) = ϕ ( e ) = ϕ ( e e ) = ϕ ( e ) ϕ ( e )
ϕ ( e ) = e
ϕ ( g 1 ) ϕ ( g ) = ϕ ( g 1 g ) = ϕ ( e ) = e
ϕ ( H 1 )
G 2
ϕ ( H 1 )
H 1
G 1
x
y
ϕ ( H 1 )
a , b H 1
ϕ ( a ) = x
ϕ ( b ) = y
x y 1 = ϕ ( a ) [ ϕ ( b ) ] 1 = ϕ ( a b 1 ) ϕ ( H 1 )
ϕ ( H 1 )
G 2
H 2
G 2
H 1
ϕ 1 ( H 2 )
H 1
g G 1
ϕ ( g ) H 2
H 1
ϕ ( e ) = e
a
b
H 1
ϕ ( a b 1 ) = ϕ ( a ) [ ϕ ( b ) ] 1
H 2
H 2
G 2
a b 1 H 1
H 1
G 1
H 2
G 2
g 1 h g H 1
h H 1
g G 1
ϕ ( g 1 h g ) = [ ϕ ( g ) ] 1 ϕ ( h ) ϕ ( g ) H 2
H 2
G 2
g 1 h g H 1
ϕ : G H
e
H
ϕ 1 ( { e } )
G
ϕ
ker ϕ
ϕ
G
H
ϕ : G H
ϕ
G
ϕ : G L 2 ( R ) R
A det ( A )
1
R
2 × 2
ker ϕ = S L 2 ( R )
ϕ : R C
ϕ ( θ ) = cos θ + i sin θ
{ 2 π n : n Z }
ker ϕ Z
ϕ
Z 7
Z 12
ϕ
Z 7
{ 0 }
Z 7
Z 7
Z 12
Z 12
7
Z 7
Z 12
G
g G
ϕ
Z
G
ϕ ( n ) = g n
g
{ 0 }
ϕ
Z
G
g
g
n
ϕ
n Z
ϕ : G H
G
ker ϕ
G
H
G
ϕ : G G / H
ϕ ( g ) = g H
ϕ ( g 1 g 2 ) = g 1 g 2 H = g 1 H g 2 H = ϕ ( g 1 ) ϕ ( g 2 )
H
ψ : G H
K = ker ψ
K
G
ϕ : G G / K
η : G / K ψ ( G )
ψ = η ϕ
K
G
η : G / K ψ ( G )
η ( g K ) = ψ ( g )
η
g 1 K = g 2 K
k K
g 1 k = g 2
η ( g 1 K ) = ψ ( g 1 ) = ψ ( g 1 ) ψ ( k ) = ψ ( g 1 k ) = ψ ( g 2 ) = η ( g 2 K )
η
η : G / K ψ ( G )
ψ = η ϕ
η
η ( g 1 K g 2 K ) = η ( g 1 g 2 K ) = ψ ( g 1 g 2 ) = ψ ( g 1 ) ψ ( g 2 ) = η ( g 1 K ) η ( g 2 K )
η
ψ ( G )
η
η ( g 1 K ) = η ( g 2 K )
ψ ( g 1 ) = ψ ( g 2 )
ψ ( g 1 1 g 2 ) = e
g 1 1 g 2
ψ
g 1 1 g 2 K = K
g 1 K = g 2 K
ψ = η ϕ
G
g
ϕ : Z G
n g n
ϕ ( m + n ) = g m + n = g m g n = ϕ ( m ) ϕ ( n )
ϕ
| g | = m
g m = e
ker ϕ = m Z
Z / ker ϕ = Z / m Z G
g
ker ϕ = 0
ϕ
G
Z
Z
Z n
H
G
G
N
G
H N
G
H N
H
H / H N H N / N
H N = { h n : h H , n N }
G
h 1 n 1 , h 2 n 2 H N
N
( h 2 ) 1 n 1 h 2 N
( h 1 n 1 ) ( h 2 n 2 ) = h 1 h 2 ( ( h 2 ) 1 n 1 h 2 ) n 2
H N
h n H N
H N
( h n ) 1 = n 1 h 1 = h 1 ( h n 1 h 1 )
H N
H
h H
n H N
h 1 n h H
H
h 1 n h N
N
G
h 1 n h H N
ϕ
H
H N / N
h h N
ϕ
h n N = h N
h
H
ϕ
ϕ ( h h ) = h h N = h N h N = ϕ ( h ) ϕ ( h )
ϕ
H / ker ϕ
H N / N = ϕ ( H ) H / ker ϕ
ker ϕ = { h H : h N } = H N
H N / N = ϕ ( H ) H / H N
N
G
H H / N
H
N
G / N
G
N
G / N
H
G
N
N
H
H / N
a N
b N
H / N
( a N ) ( b 1 N ) = a b 1 N H / N
H / N
G / N
S
G / N
N
H = { g G : g N S }
h 1 , h 2 H
( h 1 N ) ( h 2 N ) = h 1 h 2 N S
h 1 1 N S
H
G
H
N
S = H / N
H H / N
H 1
H 2
G
N
H 1 / N = H 2 / N
h 1 H 1
h 1 N H 1 / N
h 1 N = h 2 N H 2
h 2
H 2
N
H 2
h 1 H 2
H 1 H 2
H 2 H 1
H 1 = H 2
H H / N
H
G
N
H
G / N G / H
g N g H
H / N
H / N
G / N
H / N
G / N
G G / N G / N H / N
H
H
G
G
N
H
G
N H
G / H G / N H / N
Z / m Z ( Z / m n Z ) / ( m Z / m n Z )
| Z / m n Z | = m n
| Z / m Z | = m
| m Z / m n Z | = n
ϕ : Z 10 Z 10
ϕ ( x ) = x + x
ϕ
ϕ
ϕ
det ( A B ) = det ( A ) det ( B )
A , B G L 2 ( R )
G L 2 ( R )
R
ϕ : R G L 2 ( R )
ϕ ( a ) = ( 1 0 0 a )
ϕ : R G L 2 ( R )
ϕ ( a ) = ( 1 0 a 1 )
ϕ : G L 2 ( R ) R
ϕ ( ( a b c d ) ) = a + d
ϕ : G L 2 ( R ) R
ϕ ( ( a b c d ) ) = a d b c
ϕ : M 2 ( R ) R
ϕ ( ( a b c d ) ) = b
M 2 ( R )
2 × 2
R
{ 1 }
A
m × n
x A x
ϕ : R n R m
ϕ : Z Z
ϕ ( n ) = 7 n
ϕ
ϕ
ϕ ( m + n ) = 7 ( m + n ) = 7 m + 7 n = ϕ ( m ) + ϕ ( n )
ϕ
Z 24
Z 18
ϕ : Z 24 Z 18
ϕ
Z 24
ϕ
Z 18
Z
Z 12
Z 24
H = 4
N = 6
H N
H + N
H N
H N / N
H / ( H N )
H N / N
H / ( H N )
G
n N
ϕ : G G
g g n
ϕ : G H
G
ϕ ( G )
a , b G
ϕ ( a ) ϕ ( b ) = ϕ ( a b ) = ϕ ( b a ) = ϕ ( b ) ϕ ( a )
ϕ : G H
G
ϕ ( G )
G
H
k
H
G
Q / Z Q
G
N
G
H
G / N
ϕ 1 ( H )
G
| H | | N |
ϕ : G G / N
G 1
G 2
H 1
H 2
G 1
G 2
ϕ : G 1 G 2
ϕ
ϕ ¯ : ( G 1 / H 1 ) ( G 2 / H 2 )
ϕ ( H 1 ) H 2
H
K
G
H K = { e }
G
G / H × G / K
ϕ : G 1 G 2
H 1
G 1
ϕ ( H 1 ) = H 2
G 1 / H 1 G 2 / H 2
ϕ : G H
ϕ
ϕ 1 ( e ) = { e }
ϕ : G H
G
a b
ϕ ( a ) = ϕ ( b )
a , b G
Aut ( G )
G
G
G
Aut ( G ) S G
G
i g : G G
i g ( x ) = g x g 1
g G
i g Aut ( G )
Inn ( G )
Inn ( G )
Aut ( G )
G
G
i g
G
G Aut ( G )
g i g
Inn ( G )
Z ( G )
G / Z ( G ) Inn ( G )
Aut ( S 3 )
Inn ( S 3 )
D 4
ϕ : Z Z
Aut ( Z )
Z 8
Aut ( Z 8 ) U ( 8 )
k Z n
ϕ k : Z n Z n
a k a
ϕ k
ϕ k
k
Z n
Z n
ϕ k
k
Z n
ψ : U ( n ) Aut ( Z n )
ψ : k ϕ k
12
20
G = { a i | a 12 = e } H = { x i | x 20 = e }
G
ϕ : G H , ϕ ( a ) = x 5 ϕ ( a i ) = ϕ ( a ) i = ( x 5 ) i = x 5 i
4
3
12
4
20
D 20
20
5
D 20
D 5
5
72
7
4
G × H
G
H
G × H
x x
12
12
G
H
G × H
S 7
( 1 , 2 , 3 )
( 4 , 5 , 6 , 7 )
S 12
( 1 , 2 , 3 ) ( 4 , 5 , 6 ) ( 7 , 8 , 9 ) ( 10 , 11 , 12 )
( 1 , 10 , 7 , 4 ) ( 2 , 11 , 8 , 5 ) ( 3 , 12 , 9 , 6 )
S n
Z 2
S n
S 6
Z 2
2
2
S 6
D 20
D 20
D 20
D 20
D 20
20
T : R n R m
x
y
R n
α R
T ( x + y ) = T ( x ) + T ( y ) T ( α y ) = α T ( y )
m × n
R
R n
R m
x = ( x 1 , , x n ) t
y = ( y 1 , , y n ) t
R n
m × n
A = ( a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n )
R m
α
A ( x + y ) = A x + A y and α A x = A ( α x )
x = ( x 1 x 2 x n )
A
( a i j )
T : R n R m
A
T
T
e 1 = ( 1 , 0 , , 0 ) t e 2 = ( 0 , 1 , , 0 ) t e n = ( 0 , 0 , , 1 ) t
x = ( x 1 , , x n ) t
x 1 e 1 + x 2 e 2 + + x n e n
T ( e 1 ) = ( a 11 , a 21 , , a m 1 ) t , T ( e 2 ) = ( a 12 , a 22 , , a m 2 ) t , T ( e n ) = ( a 1 n , a 2 n , , a m n ) t
T ( x ) = T ( x 1 e 1 + x 2 e 2 + + x n e n ) = x 1 T ( e 1 ) + x 2 T ( e 2 ) + + x n T ( e n ) = ( k = 1 n a 1 k x k , , k = 1 n a m k x k ) t = A x
T : R 2 R 2
T ( x 1 , x 2 ) = ( 2 x 1 + 5 x 2 , 4 x 1 + 3 x 2 )
T
T e 1 = ( 2 , 4 ) t
T e 2 = ( 5 , 3 ) t
T
A = ( 2 5 4 3 )
n × n
A
A 1
A A 1 = A 1 A = I
I = ( 1 0 0 0 1 0 0 0 1 )
n × n
A
A
A
( 2 1 5 3 )
A
A 1 = ( 3 1 5 2 )
A 1
det ( A ) = 2 3 5 1 = 1
A
B
n × n
det ( A B ) = ( det A ) ( det B )
A
det ( A 1 ) = 1 / det A
A = ( a i j )
A t = ( a j i )
det ( A t ) = det A
T
n × n
A
T
| det A |
R 2
T
| det A |
2 × 2
n × n
G L n ( R )
det ( A ) = 1
det ( B ) = 1
det ( A B ) = det ( A ) det ( B ) = 1
det ( A 1 ) = 1 / det A = 1
S L n ( R )
2 × 2
A = ( a b c d )
A
a d b c
G L 2 ( R )
a d b c   0
A
A 1 = 1 a d b c ( d b c a )
A
S L 2 ( R )
A 1 = ( d b c a )
S L 2 ( R )
A = ( 1 1 0 1 )
S L 2 ( R )
x = ( 1 , 0 ) t
y = ( 0 , 1 ) t
A
( 1 , 0 ) t
( 1 , 1 ) t
A x = ( 1 , 0 ) t
A y = ( 1 , 1 ) t
S L 2 ( R )
O ( n )
G L n ( R )
A
A 1 = A t
O ( n )
n × n
O ( n )
G L n ( R )
( 3 / 5 4 / 5 4 / 5 3 / 5 ) , ( 1 / 2 3 / 2 3 / 2 1 / 2 ) , ( 1 / 2 0 1 / 2 1 / 6 2 / 6 1 / 6 1 / 3 1 / 3 1 / 3 )
O ( n )
x = ( x 1 , , x n ) t
y = ( y 1 , , y n ) t
x , y = x t y = ( x 1 , x 2 , , x n ) ( y 1 y 2 y n ) = x 1 y 1 + + x n y n
x = ( x 1 , , x n ) t
x
x = x , x = x 1 2 + + x n 2
x
y
x y
x
y
w
R n
α R
x , y = y , x
x , y + w = x , y + x , w
α x , y = x , α y = α x , y
x , x 0
x = 0
x , y = 0
x
R n
y = 0
x = ( 3 , 4 ) t
3 2 + 4 2 = 5
A = ( 3 / 5 4 / 5 4 / 5 3 / 5 )
A x = ( 7 / 5 , 24 / 5 ) t
det ( A A t ) = det ( I ) = 1
det ( A ) = det ( A t )
1
1
a j = ( a 1 j a 2 j a n j )
A = ( a i j )
A A t = I
a r , a s = δ r s
δ r s = { 1 r = s 0 r   s
n × n
A
A 1 = A t
A
A x A y = x y
A x = x
A x , A y = x , y
A
n × n
A
A 1 = A t
x
y
A x , A y = x , y
x
y
A x A y = x y
x
A x = x
( 2 ) ( 3 )
A x , A y = ( A x ) t A y = x t A t A y = x t y = x , y
( 3 ) ( 2 )
x , x = A x , A x = x t A t A x = x , A t A x
x , ( A t A I ) x = 0
x
A t A I = 0
A 1 = A t
( 3 ) ( 4 )
A
A
A x A y 2 = A ( x y ) 2 = A ( x y ) , A ( x y ) = x y , x y = x y 2
( 4 ) ( 5 )
A
A
y = 0
A x = A x A y = x y = x
( 5 ) ( 3 )
x , y = 1 2 [ x + y 2 x 2 y 2 ]
A x , A y = 1 2 [ A x + A y 2 A x 2 A y 2 ] = 1 2 [ A ( x + y ) 2 A x 2 A y 2 ] = 1 2 [ x + y 2 x 2 y 2 ] = x , y
O ( 2 )
R 2
R 2
A O ( 2 )
e 1 = ( 1 , 0 ) t
e 2 = ( 0 , 1 ) t
A e 1 = ( a , b ) t
a 2 + b 2 = 1
A
O ( 2 )
A e 2 = ± ( b , a ) t
A e 2 = ( b , a ) t
A = ( a b b a ) = ( cos θ sin θ sin θ cos θ ) ,
0 θ < 2 π
A
R 2
θ
A e 2 = ( b , a ) t
B = ( a b b a ) = ( cos θ sin θ sin θ cos θ ) .
det B = 1
B 2 = ( 1 0 0 1 ) .
C = ( 1 0 0 1 ) ,
B = A C
S O ( n )
O ( n )
S L n ( R )
O ( n )
E ( n )
( A , x )
A
O ( n )
x
R n
( A , x ) ( B , y ) = ( A B , A y + x )
( I , 0 )
( A , x )
( A 1 , A 1 x )
E ( n )
R 2
R n
f
R n
R n
f
f ( x ) f ( y ) = x y
x , y R n
f
O ( n )
R n
O ( n )
R n
x
T y ( x ) = x + y
T
O ( n )
R 2
R 2
R n
f
R 2
f
O ( 2 )
f
R 2
f
f ( 0 ) = 0
f ( x ) = x
x 2 2 f ( x ) , f ( y ) + y 2 = f ( x ) 2 2 f ( x ) , f ( y ) + f ( y ) 2 = f ( x ) f ( y ) , f ( x ) f ( y ) = f ( x ) f ( y ) 2 = x y 2 = x y , x y = x 2 2 x , y + y 2
f ( x ) , f ( y ) = x , y
e 1
e 2
( 1 , 0 ) t
( 0 , 1 ) t
x = ( x 1 , x 2 ) = x 1 e 1 + x 2 e 2
f ( x ) = f ( x ) , f ( e 1 ) f ( e 1 ) + f ( x ) , f ( e 2 ) f ( e 2 ) = x 1 f ( e 1 ) + x 2 f ( e 2 )
f
f
T x f
x
R 2
T x f ( y ) = A y
A O ( 2 )
f ( y ) = A y + x
f ( y ) = A y + x 1 g ( y ) = B y + x 2
f ( g ( y ) ) = f ( B y + x 2 ) = A B y + A x 2 + x 1
R 2
E ( 2 )
R 2
E ( 2 )
R n
R n
X R n
X
R n
X
R 1
Z 2
R 2
O ( 2 )
R 2
Z n
D n
G
E ( 2 )
G
R 2
O ( 2 )
O ( 2 )
R θ = ( cos θ sin θ sin θ cos θ )
T ϕ = ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( 1 0 0 1 ) = ( cos ϕ sin ϕ sin ϕ cos ϕ )
det ( R θ ) = 1
det ( T ϕ ) = 1
T ϕ 2 = I
G
G
1
G
G
G
θ 0
R θ 0
R θ 0
G
n
θ 1
n θ 0
( n + 1 ) θ 0
( n + 1 ) θ 0 θ 1
θ 0
θ 0
G
T
ϕ : G { 1 , 1 }
A det ( A )
| G / ker ϕ | = 2
G
n
| G | = 2 n
G
R θ , , R θ n 1 , T R θ , , T R θ n 1
T R θ T = R θ 1
G
D n
R 3
R 2
R 3
x
y
R 2
x
y
m x + n y
m
n
x
y
( 1 , 1 ) t
( 2 , 0 ) t
( 1 , 1 ) t
( 1 , 1 ) t
{ x 1 , x 2 }
{ y 1 , y 2 }
y 1 = α 1 x 1 + α 2 x 2 y 2 = β 1 x 1 + β 2 x 2
α 1
α 2
β 1
β 2
U = ( α 1 α 2 β 1 β 2 )
x 1
x 2
y 1
y 2
U 1
U 1 ( y 1 y 2 ) = ( x 1 x 2 )
U
U 1
U
U 1
U U 1 = I
det ( U U 1 ) = det ( U ) det ( U 1 ) = 1 ;
det ( U ) = ± 1
± 1
( 3 1 5 2 )
R 2
E ( 2 )
R 2
R 3
G E ( 2 )
{ ( I , t ) : t L }
L
R 2
Z × Z
G
G 0 = { A : ( A , b ) G for some b }
G 0
O ( 2 )
x
L
G
H
G 0
( A , y )
G
( A , y ) ( I , x ) ( A , y ) 1 = ( A , A x + y ) ( A 1 , A 1 y ) = ( A A 1 , A A 1 y + A x + y ) = ( I , A x ) ;
( I , A x )
G
A x
L
G 0
G
T
G
G / T G 0
Z n
D n
n = 1 , 2 , 3 , 4 , 6
R 2
Z 1
Z 2
Z 3
Z 4
Z 6
D 1
D 1
D 1
D 2
D 2
D 2
D 2
D 3
D 4
D 6
n
R 3
R 4
R 5
R 3
2 × 2
R n
R 2
x , y = 1 2 [ x + y 2 x 2 y 2 ]
1 2 [ x + y 2 + x 2 y 2 ] = 1 2 [ x + y , x + y x 2 y 2 ] = 1 2 [ x 2 + 2 x , y + y 2 x 2 y 2 ] = x , y
O ( n )
S O ( n )
( 1 / 2 1 / 2 1 / 2 1 / 2 )
( 1 / 5 2 / 5 2 / 5 1 / 5 )
( 4 / 5 0 3 / 5 3 / 5 0 4 / 5 0 1 0 )
( 1 / 3 2 / 3 2 / 3 2 / 3 2 / 3 1 / 3 2 / 3 1 / 3 2 / 3 )
S O ( 2 )
O ( 3 )
x
y
w
R n
α R
x , y = y , x
x , y + w = x , y + x , w
α x , y = x , α y = α x , y
x , x 0
x = 0
x , y = 0
x
R n
y = 0
x , y = y , x
E ( n ) = { ( A , x ) : A O ( n ) and x R n }
{ ( 2 , 1 ) , ( 1 , 1 ) }
{ ( 12 , 5 ) , ( 7 , 3 ) }
( 5 2 2 1 )
G
E ( 2 )
T
G
G
G / T
A S L 2 ( R )
x
y
R 2
A x
A y
S O ( n )
O ( n )
det : O ( n ) R
S O ( n )
f
R n
E ( 2 )
( A , x )
x   0
O ( n )
x = ( x 1 , x 2 )
R 2
x 1 2 + x 2 2 = 1
A O ( 2 )
A x
G
H
N
G
N
H
H N = { i d }
H N = G
S 3
A 3
H = { ( 1 ) , ( 12 ) }
Q 8
E ( 2 )
O ( 2 )
H
H
R 2
p 6 m
n
Z n
G = H n H n 1 H 1 H 0 = { e }
H i
H i + 1
H i + 1 / H i
G
Z p
p
Z m n Z m × Z n
gcd ( m , n ) = 1
Z p 1 α 1 × × Z p n α n
p k
G
{ g i }
G
i
I
G
g i
G
g i
G
G
G
{ g i : i I }
g i
G
{ g i : i I }
G
G
S 3
( 12 )
( 123 )
Z × Z n
{ ( 1 , 0 ) , ( 0 , 1 ) }
Q
Q
p 1 / q 1 , , p n / q n
p i / q i
p
q 1 , , q n
1 / p
Q
p 1 / q 1 , , p n / q n
p
p i / q i + p j / q j = ( p i q j + p j q i ) / ( q i q j )
H
G
{ g i G : i I }
h H
h = g i 1 α 1 g i n α n
g i k
K
g i 1 α 1 g i n α n
g i k
K
H
K
G
K = H
H
g i
K
g i 0 = 1
K
g = g i 1 k 1 g i n k n
K
K
g 1 = ( g i 1 k 1 g i n k n ) 1 = ( g i n k n g i 1 k 1 )
g i
g i
a 3 b 5 a 7
a 4 b 5
p
G
p
p
G
p
Z 2 × Z 2
Z 4
2
Z 27
3
p
G
Z p 1 α 1 × Z p 2 α 2 × × Z p n α n
p i
540 = 2 2 3 3 5
Z 2 × Z 2 × Z 3 × Z 3 × Z 3 × Z 5
Z 2 × Z 2 × Z 3 × Z 9 × Z 5
Z 2 × Z 2 × Z 27 × Z 5
Z 4 × Z 3 × Z 3 × Z 3 × Z 5
Z 4 × Z 3 × Z 9 × Z 5
Z 4 × Z 27 × Z 5
G
n
p
n
G
p
n = 1
k
k < n
p
n
G
G = a
a
G
p
n
p = n
G
p 1
p
G
H
1 < | H | < n
p | H |
H
p
p
H
G
H
G
| G | = | H | | G / H |
p
| G / H |
| G / H | < | G | = n
G / H
a H
p
H = ( a H ) p = a p H
a p H
a H
| H | = r
p
r
s
t
s p + t r = 1
a p
r
( a p ) r = ( a r ) p = 1
a r
p
a r   1
a r = 1
a = a s p + t r = a s p a t r = ( a p ) s ( a r ) t = ( a p ) s 1 = ( a p ) s
a p H
a = ( a p ) s H
a r   1
p
G
G
p
p
G
G
p
p
p
| G | = p n
g G
p n
p
| G |
p
q
G
q
p
G
n = p 1 α 1 p k α k
p 1 , , p k
α 1 , α 2 , , α k
G
G 1 , G 2 , , G k
G i
G
p i r
r
G
G i
G
i = 1 , , k
p i 0 = 1
1 G i
g G i
p i r
g 1
p i r
h G i
p i s
( g h ) p i t = g p i t h p i t = 1 1 = 1
t
r
s
G = G 1 G 2 G k
G i G j = { 1 }
i   j
g 1 G 1
G 2 , G 3 , , G k
g 1 = g 2 g 3 g k
g i G i
g i
p α i
g i p α i = 1
i = 2 , 3 , , k
g 1 p 2 α 2 p k α k = 1
g 1
p 1
gcd ( p 1 , p 2 α 2 p k α k ) = 1
g 1 = 1
G 1
G 2 , G 3 , , G k
G i G j = { 1 }
i   j
g G
g 1 g k
g i G i
g
G
| g | = p 1 β 1 p 2 β 2 p k β k
β 1 , , β k
a i = | g | / p i β i
a i
b 1 , , b k
a 1 b 1 + + a k b k = 1
g = g a 1 b 1 + + a k b k = g a 1 b 1 g a k b k
g ( a i b i ) p i β i = g b i | g | = e
g a i b i
G i
g i = g a i b i
g = g 1 g k G 1 G 2 G k
G = G 1 G 2 G k
p i
G i
G
p
g G
G
g × H
H
G
G
p n
n
n = 1
G
p
g
k
1 k < n
g
G
| g | = p m
a p m = e
a G
h
G
h g
h
h
G = g
H = h
g H = { e }
| H | = p
| h p | = | h | / p
h p
h
g
h
h p = g r
r
( g r ) p m 1 = ( h p ) p m 1 = h p m = e
g r
p m 1
g r
g
p
r
r = p s
h p = g r = g p s
a
g s h
a
g
h
g
a p = g s p h p = g r h p = h p h p = e
a
p
a g
h
g
| H | = p
g H
G / H
g
G
| g H | < | g | = p m
H = ( g H ) p m 1 = g p m 1 H ;
g p m 1
g H = { e }
g
p m
g H
G / H
G / H g H × K / H
K
G
H
g K = { e }
b g K
b H g H K / H = { H }
b g H = { e }
G = g K
G g × K
G
g
G
g = G
G Z | g | × H
H
G
| H | < | G |
G
Z p 1 α 1 × Z p 2 α 2 × × Z p n α n × Z × × Z
p i
G
G = H n H n 1 H 1 H 0 = { e }
H i
H i + 1
H i
G
Z 9 Z 45 Z 180 Z { 0 } , Z 24 2 6 12 { 0 }
D 4
D 4 { ( 1 ) , ( 12 ) ( 34 ) , ( 13 ) ( 24 ) , ( 14 ) ( 23 ) } { ( 1 ) , ( 12 ) ( 34 ) } { ( 1 ) }
{ ( 1 ) , ( 12 ) ( 34 ) }
D 4
{ K j }
{ H i }
{ H i } { K j }
H i
K j
Z 3 Z 9 Z 45 Z 90 Z 180 Z { 0 }
Z 9 Z 45 Z 180 Z { 0 }
{ H i }
G
H i + 1 / H i
{ H i }
{ K j }
G
{ H i + 1 / H i }
{ K j + 1 / K j }
Z 60 3 15 { 0 } Z 60 4 20 { 0 }
Z 60
Z 60 / 3 20 / { 0 } Z 3 3 / 15 4 / 20 Z 5 15 / { 0 } Z 60 / 4 Z 4
{ H i }
G
{ H i }
G
Z 60
Z 60 3 15 30 { 0 }
Z 60 / 3 Z 3 3 / 15 Z 5 15 / 30 Z 2 30 / { 0 } Z 2
Z 60
Z 60 2 4 20 { 0 }
n 5
S n A n { ( 1 ) }
S n
S n / A n Z 2
A n
{ 0 } = H 0 H 1 H n 1 H n = Z
H 1
k Z
k N
H 1 / H 0 k Z
Z 60
Z 60
Z 2
Z 2
Z 3
Z 5
G
G
k
1 k < n
G = H n H n 1 H 1 H 0 = { e } G = K m K m 1 K 1 K 0 = { e }
G
G
H i K m 1
H i + 1 K m 1
K j H n 1
K j + 1 H n 1
G = H n H n 1 H n 1 K m 1 H 0 K m 1 = { e } G = K m K m 1 K m 1 H n 1 K 0 H n 1 = { e }
H i K m 1
H i + 1 K m 1
( H i + 1 K m 1 ) / ( H i K m 1 ) = ( H i + 1 K m 1 ) / ( H i ( H i + 1 K m 1 ) ) H i ( H i + 1 K m 1 ) / H i
H i
H i ( H i + 1 K m 1 )
{ H i }
H i + 1 / H i
H i ( H i + 1 K m 1 ) / H i
H i + 1 / H i
H i / H i
H i ( H i + 1 K m 1 )
H i
H i + 1
H n 1 H n 1 K m 1 H 0 K m 1 = { e }
H n 1
H n 1 H 1 H 0 = { e }
G = H n H n 1 H 1 H 0 = { e }
G = H n H n 1 H n 1 K m 1 H 0 K m 1 = { e }
H n 1 = K m 1
{ H i }
{ K j }
H n 1 K m 1
G
H n 1
H n 1 K m 1 = G
K m 1 / ( K m 1 H n 1 ) ( H n 1 K m 1 ) / H n 1 = G / H n 1
G = H n H n 1 H n 1 K m 1 H 0 K m 1 = { e }
G = K m K m 1 K m 1 H n 1 K 0 H n 1 = { e }
G
{ H i }
H i + 1 / H i
S 4
S 4 A 4 { ( 1 ) , ( 12 ) ( 34 ) , ( 13 ) ( 24 ) , ( 14 ) ( 23 ) } { ( 1 ) }
n 5
S n A n { ( 1 ) }
S n
S n
n 5
16
200 = 2 3 5 2
729 = 3 6
Z 8 × Z 3 × Z 3
72
8
G
40
200
720
Z 12
Z 48
Q 8
D 4
S 3 × Z 4
S 4
S n
n 5
Q
{ 0 } 6 3 Z 12
{ ( 1 ) } × { 0 } { ( 1 ) , ( 123 ) , ( 132 ) } × { 0 } S 3 × { 0 } S 3 × 2 S 3 × Z 4
G = Z 2 × Z 2 ×
G
m
n
m
G
n
G
G
G
H
K
G × H G × K
H K
G
H
G × H
G
N
G
N
N
G
N
G / N
G
N
G
N
G / N
G
N
G / N
N = N n N n 1 N 1 N 0 = { e } G / N = G n / N G n 1 / N G 1 / N G 0 / N = { N }
G
G
G = P n P n 1 P 1 P 0 = { e }
P i
P i + 1
P i + 1 / P i
G
G
G
N
G
G / N
D n
n
D n
2
G
N
G
N
G / N
G
p
H
K
H
K
K
H
G
n 2
G
G
G
G
a 1 b 1 a b
a , b G
G
G ( 0 ) = G
G ( 1 ) = G
G ( i + 1 ) = ( G ( i ) )
G ( i + 1 )
( G ( i ) )
G ( 0 ) = G G ( 1 ) G ( 2 )
G
G
G ( n ) = { e }
n
G
n 2
G
G / G
H
K
G
H
K
H
K
H ( H K )
H ( H K )
K ( H K )
K ( H K )
H ( H K ) / H ( H K ) K ( H K ) / K ( H K ) ( H K ) / ( H K ) ( H K )
G
n
n
16
2 p
p
2 p
2 p
p
n = 6 , 10 , 14
n = 9
p 2
p
Z 9
Z 3 × Z 3
n = 15
Z 3 × Z 5 Z 15
n = 8
n = 12
n = 16
14
n = 8
3
n = 12
2
3
4
4
12
8
2 k
k > 2
16
16
Z 3 Z 4
G
X
g G
x X
g x
X
X
G
G
X
G × X X
( g , x ) g x
e x = x
x X
( g 1 g 2 ) x = g 1 ( g 2 x )
x X
g 1 , g 2 G
X
G
G
X
G
G
X
( g , x ) x
X
G
G = G L 2 ( R )
X = R 2
G
X
v R 2
I
I v = v
A
B
2 × 2
( A B ) v = A ( B v )
G = D 4
X = { 1 , 2 , 3 , 4 }
D 4
{ ( 1 ) , ( 13 ) , ( 24 ) , ( 1432 ) , ( 1234 ) , ( 12 ) ( 34 ) , ( 14 ) ( 23 ) , ( 13 ) ( 24 ) }
D 4
X
( 13 ) ( 24 )
1
3
2
4
X
G
S X
X
X
G
( σ , x ) σ ( x )
σ G
x X
X = G
G
( g , x ) λ g ( x ) = g x
λ g
e x = λ e x = e x = x ( g h ) x = λ g h x = λ g λ h x = λ g ( h x ) = g ( h x )
H
G
G
H
H
G
X = G
H
G
G
H
H
G
H × G G
( h , g ) h g h 1
h H
g G
( h 1 h 2 , g ) = h 1 h 2 g ( h 1 h 2 ) 1 = h 1 ( h 2 g h 2 1 ) h 1 1 = ( h 1 , ( h 2 , g ) )
H
G
L H
H
L H
G
( g , x H ) g x H
( g g ) x H = g ( g x H )
G
X
x , y X
x
G
G
y
g G
g x = y
x G y
x y
G
X
G
G
X
e x = x
x y
x , y X
g
g x = y
g 1 y = x
y x
x y
y z
g
h
g x = y
h y = z
z = h y = ( h g ) x
x
z
X
G
X
G
X
G
x
X
O x
x
G
G = { ( 1 ) , ( 1 2 3 ) , ( 1 3 2 ) , ( 4 5 ) , ( 1 2 3 ) ( 4 5 ) , ( 1 3 2 ) ( 4 5 ) }
X = { 1 , 2 , 3 , 4 , 5 }
X
G
O 1 = O 2 = O 3 = { 1 , 2 , 3 }
O 4 = O 5 = { 4 , 5 }
G
X
g
G
g
X
X g
x X
g x = x
g
g
x X
G
x
x
G x
x
X g X
G x G
X = { 1 , 2 , 3 , 4 , 5 , 6 }
G
{ ( 1 ) , ( 1 2 ) ( 3 4 5 6 ) , ( 3 5 ) ( 4 6 ) , ( 1 2 ) ( 3 6 5 4 ) }
X
G
X ( 1 ) = X , X ( 3 5 ) ( 4 6 ) = { 1 , 2 } , X ( 1 2 ) ( 3 4 5 6 ) = X ( 1 2 ) ( 3 6 5 4 ) =
G 1 = G 2 = { ( 1 ) , ( 3 5 ) ( 4 6 ) } , G 3 = G 4 = G 5 = G 6 = { ( 1 ) }
G x
G
x X
G
X
x X
x
G x
G
e G x
X
g , h G x
g x = x
h x = x
( g h ) x = g ( h x ) = g x = x
G x
G x
g G x
x = e x = ( g 1 g ) x = ( g 1 ) g x = g 1 x
g 1
G x
g G
| X g |
x X
| O x |
x X
G x
G
G
X
G
x X
| O x | = [ G : G x ]
| G | / | G x |
G x
G
ϕ
O x
X
L G x
G x
G
y O x
g
G
g x = y
ϕ
ϕ ( y ) = g G x
ϕ
ϕ ( y 1 ) = ϕ ( y 2 )
ϕ ( y 1 ) = g 1 G x = g 2 G x = ϕ ( y 2 )
g 1 x = y 1
g 2 x = y 2
g 1 G x = g 2 G x
g G x
g 2 = g 1 g
y 2 = g 2 x = g 1 g x = g 1 x = y 1 ;
ϕ
ϕ
g G x
g x = y
ϕ ( y ) = g G x
X
G
X G
X
X G = { x X : g x = x for all g G }
X
| X | = | X G | + i = k n | O x i |
x k , , x n
X
G
( g , x ) g x g 1
G
Z ( G ) = { x : x g = g x for all g G }
G
x 1 , , x k
G
| O x 1 | = n 1 , , | O x k | = n k
| G | = | Z ( G ) | + n 1 + + n k
x i
C ( x i ) = { g G : g x i = x i g }
x i
| G | = | Z ( G ) | + [ G : C ( x 1 ) ] + + [ G : C ( x k ) ]
G
S 3
{ ( 1 ) } , { ( 123 ) , ( 132 ) } , { ( 12 ) , ( 13 ) , ( 23 ) }
6 = 1 + 2 + 3
D 4
{ ( 1 ) , ( 13 ) ( 24 ) }
{ ( 13 ) , ( 24 ) } , { ( 1432 ) , ( 1234 ) } , { ( 12 ) ( 34 ) , ( 14 ) ( 23 ) }
D 4
8 = 2 + 2 + 2 + 2
S n
σ = ( a 1 , , a k )
τ S n
τ σ τ 1 = ( τ ( a 1 ) , , τ ( a k ) )
σ = σ 1 σ 2 σ r
σ i
r i
σ
τ S n
S n
n
S 3
3
3 = 1 + 1 + 1 3 = 1 + 2 3 = 3 ;
n
n
G
p n
p
G
| G | = | Z ( G ) | + n 1 + + n k
n i > 1
n i | G |
p
n i
p | G |
p
| Z ( G ) |
G
| Z ( G ) | 1
| Z ( G ) | p
g Z ( G )
g   1
G
p 2
p
G
| Z ( G ) | = p
p 2
| Z ( G ) | = p 2
| Z ( G ) | = p
Z ( G )
G / Z ( G )
p
a Z ( G )
G / Z ( G )
g Z ( G )
a m Z ( G )
m
g = a m x
x
G
h Z ( G ) G / Z ( G )
y
Z ( G )
h = a n y
n
x
y
G
G
g h = a m x a n y = a m + n x y = a n y a m x = h g
G
2 4 = 16
90
X
G
x y
G x
G y
| G x | = | G y |
G
X
( g , x ) g x
x y
g G
g x = y
a G x
g a g 1 y = g a g 1 y = g a x = g x = y
ϕ : G x G y
ϕ ( a ) = g a g 1
ϕ
ϕ ( a b ) = g a b g 1 = g a g 1 g b g 1 = ϕ ( a ) ϕ ( b )
ϕ ( a ) = ϕ ( b )
g a g 1 = g b g 1
a = b
ϕ
b
G y
g 1 b g
G x
g 1 b g x = g 1 b g x = g 1 b y = g 1 y = x ;
ϕ ( g 1 b g ) = b
G
X
k
X
k = 1 | G | g G | X g |
x
g G
g
x
g x = x
g
x
g G | X g |
x X | G x | ;
g G | X g | = x X | G x |
y O x | G y | = | O x | | G x |
| G |
k
g G | X g | = x X | G x | = k | G |
X = { 1 , 2 , 3 , 4 , 5 }
G
G = { ( 1 ) , ( 1 3 ) , ( 1 3 ) ( 2 5 ) , ( 2 5 ) }
X
{ 1 , 3 }
{ 2 , 5 }
{ 4 }
X ( 1 ) = X X ( 1 3 ) = { 2 , 4 , 5 } X ( 1 3 ) ( 2 5 ) = { 4 } X ( 2 5 ) = { 1 , 3 , 4 }
k = 1 | G | g G | X g | = 1 4 ( 5 + 3 + 1 + 3 ) = 3
D 4
( 1 ) ( 13 ) ( 24 ) ( 1432 ) ( 1234 ) ( 12 ) ( 34 ) ( 14 ) ( 23 ) ( 13 ) ( 24 )
G
{ 1 , 2 , 3 , 4 }
X
Y = { B , W }
B
W
f : X Y
σ D 4
σ ~
σ ~ ( f ) = f σ
f : X Y
f
f ( 1 ) = B f ( 2 ) = W f ( 3 ) = W f ( 4 ) = W
σ = ( 1 2 ) ( 3 4 )
σ ~ ( f ) = f σ
2
B
W
σ ~
G ~
X ~
X ~
X
Y
G ~
X ~ ( 1 ) = X ~
| X ~ | = 2 4 = 16
X ~ ( 1 2 3 4 )
f X ~
f
( 1 23 4 )
f ( 1 ) = f ( 2 ) = f ( 3 ) = f ( 4 )
f
f ( x ) = B
f ( x ) = W
x
| X ~ ( 1 2 3 4 ) | = 2
| X ~ ( 1 4 3 2 ) | = 2
X ~ ( 1 3 ) ( 2 4 )
f ( 1 ) = f ( 3 )
f ( 2 ) = f ( 4 )
| X ~ ( 13 ) ( 24 ) | = 2 2 = 4
| X ~ ( 1 2 ) ( 3 4 ) | = 4
| X ~ ( 1 4 ) ( 2 3 ) | = 4
X ~ ( 1 3 )
f ( 1 ) = f ( 3 )
| X ~ ( 1 3 ) | = 2 3 = 8
| X ~ ( 2 4 ) | = 8
1 8 ( 2 4 + 2 1 + 2 2 + 2 1 + 2 2 + 2 2 + 2 3 + 2 3 ) = 6
G
X
X ~
X
Y
G ~
X ~
σ ~ G ~
σ ~ ( f ) = f σ
σ G
f X ~
n
σ
| X ~ σ | = | Y | n
σ G
f X ~
f σ
X ~
g
X
Y
σ ~ ( f ) = σ ~ ( g )
x X
f ( σ ( x ) ) = σ ~ ( f ) ( x ) = σ ~ ( g ) ( x ) = g ( σ ( x ) )
σ
X
x
X
x
X
σ
f
g
X
f = g
σ ~
σ σ ~
σ
X
σ = σ 1 σ 2 σ n
f
X ~ σ
σ
n
| Y |
| X ~ σ | = | Y | n
X = { 1 , 2 , , 7 }
Y = { A , B , C }
g
X
( 1 3 ) ( 2 4 5 ) = ( 1 3 ) ( 2 4 5 ) ( 6 ) ( 7 )
n = 4
f X ~ g
g
| Y | = 3
| X ~ g | = 3 4 = 81
1 8 ( 4 4 + 4 1 + 4 2 + 4 1 + 4 2 + 4 2 + 4 3 + 4 3 ) = 55
n
n
n
Z 2 n
Z 2
n
2 n
n
2 2 n
n
a
b
f
g
g
f
g ( a , b , c ) = f ( b , c , a )
g f
( a c b )
( a b )
f 2 f 4 f 3 f 5 f 10 f 12 f 11 f 13
f 0
f 1
f 2
f 3
f 4
f 5
f 6
f 7
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
0
1
1
1
1
0
1
0
1
0
1
0
1
f 8
f 9
f 10
f 11
f 12
f 13
f 14
f 15
0
0
1
1
1
1
1
1
1
1
0
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
0
1
1
1
1
0
1
0
1
0
1
0
1
2 2 3 = 256
2 2 4 = 65,536
a
b
c
f
g
f
g
{ a , b , c }
a
b
c
2 3
( a , b , c )
( a c b )
( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 0 ) ( 0 , 1 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 0 , 1 ) ( 1 , 1 , 1 ) ( 1 , 1 , 1 )
X
n
| X | = 2 n
( 0 , , 0 , 1 ) 0 ( 0 , , 1 , 0 ) 1 ( 0 , , 1 , 1 ) 2 ( 1 , , 1 , 1 ) 2 n 1
( a ) , ( a c ) , ( b d ) , ( a d c b ) , ( a b c d ) , ( a b ) ( c d ) , ( a d ) ( b c ) , ( a c ) ( b d )
1 8 ( 2 16 + 2 2 12 + 2 2 6 + 3 2 10 ) = 9616
( a )
( 0 )
( a c )
( 2 , 8 ) ( 3 , 9 ) ( 6 , 12 ) ( 7 , 13 )
( b d )
( 1 , 4 ) ( 3 , 6 ) ( 9 , 12 ) ( 11 , 14 )
( a d c b )
( 1 , 2 , 4 , 8 ) ( 3 , 6.12 , 9 ) ( 5 , 10 ) ( 7 , 14 , 13 , 11 )
( a b c d )
( 1 , 8 , 4 , 2 ) ( 3 , 9 , 12 , 6 ) ( 5 , 10 ) ( 7 , 11 , 13 , 14 )
( a b ) ( c d )
( 1 , 2 ) ( 4 , 8 ) ( 5 , 10 ) ( 6 , 9 ) ( 7 , 11 ) ( 13 , 14 )
( a d ) ( b c )
( 1 , 8 ) ( 2 , 4 ) ( 3 , 12 ) ( 5 , 10 ) ( 7 , 14 ) ( 11 , 13 )
( a c ) ( b d )
( 1 , 4 ) ( 2 , 8 ) ( 3 , 12 ) ( 6 , 9 ) ( 7 , 13 ) ( 11 , 14 )
G
G = H n H n 1 H 1 H 0 = { e }
H i
H i + 1
H i + 1 / H i
49
G
X
G
G
0
R 2 { 0 }
X = { 1 , 2 , 3 , 4 }
X g
G x
X = { 1 , 2 , 3 }
G = S 3 = { ( 1 ) , ( 12 ) , ( 13 ) , ( 23 ) , ( 123 ) , ( 132 ) }
X = { 1 , 2 , 3 , 4 , 5 , 6 }
G = { ( 1 ) , ( 12 ) , ( 345 ) , ( 354 ) , ( 12 ) ( 345 ) , ( 12 ) ( 354 ) }
X ( 1 ) = { 1 , 2 , 3 }
X ( 12 ) = { 3 }
X ( 13 ) = { 2 }
X ( 23 ) = { 1 }
X ( 123 ) = X ( 132 ) =
G 1 = { ( 1 ) , ( 23 ) }
G 2 = { ( 1 ) , ( 13 ) }
G 3 = { ( 1 ) , ( 12 ) }
G
X
G
x X
| G | = | O x | | G x |
O 1 = O 2 = O 3 = { 1 , 2 , 3 }
G
θ G
R 2
θ
P
R 2
G
P
G P
G = A 4
G
( g , h ) g h g 1
G
G
S 4
D 5
Z 9
Q 8
S 4
O ( 1 ) = { ( 1 ) } , O ( 12 ) = { ( 12 ) , ( 13 ) , ( 14 ) , ( 23 ) , ( 24 ) , ( 34 ) } , O ( 12 ) ( 34 ) = { ( 12 ) ( 34 ) , ( 13 ) ( 24 ) , ( 14 ) ( 23 ) } , O ( 123 ) = { ( 123 ) , ( 132 ) , ( 124 ) , ( 142 ) , ( 134 ) , ( 143 ) , ( 234 ) , ( 243 ) } , O ( 1234 ) = { ( 1234 ) , ( 1243 ) , ( 1324 ) , ( 1342 ) , ( 1423 ) , ( 1432 ) }
1 + 3 + 6 + 6 + 8 = 24
S 5
A 5
( 3 4 + 3 1 + 3 2 + 3 1 + 3 2 + 3 2 + 3 3 + 3 3 ) / 8 = 21
1 , , 6
S 4
( a b c d )
( a b ) ( c d )
( a b ) ( c d ) ( e f )
( a b c ) ( d e f )
12
( 1 2 6 + 3 2 4 + 4 2 3 + 2 2 2 + 2 2 1 ) / 12 = 13
C H 3
x 1
x 2
x 3
S 3
x 1
x 2
x 3
x 4
S 4
( 1 2 8 + 3 2 6 + 2 2 4 ) / 6 = 80
x 1
x 2
x 3
x 4
S 4
( x 1 x 2 x 3 x 4 )
12
G
X
G
X
G
X
G
X
p
p n
S n
a G
g G
g C ( a ) g 1 = C ( g a g 1 )
x g C ( a ) g 1
g 1 x g C ( a )
| G | = p n
p
| Z ( G ) | < p n 1
G
p n
p
X
G
X G = { x X : g x = x for all g G }
X
| X | | X G | ( mod p )
G
p n
p
n 2
G
p
n 3
G
p 2
S n
n
g x g 1
g 1 x g
4
4
16
0
1
24
48
4
000 , 010 , 110 , 100
001 , 011 , 111 , 101
000
6
48
8
000
100
010
001
000
3 ! = 6
11
2
2
10
11
3
S 5
A 5
7
8
D 7
D 8
Q 4
16
G
m
n
m
G
n
A 4
12
6
G
G
( g , x ) g x g 1
x 1 , , x k
G
| G | = | Z ( G ) | + [ G : C ( x 1 ) ] + + [ G : C ( x k ) ]
Z ( G ) = { g G : g x = x g for all x G }
G
C ( x i ) = { g G : g x i = x i g }
x i
p
p
G
p
p
G
p
p
G
p
p
p
G
p
p
G
G
p
G
| G | = p
G
k
p k < n
p
k
p
| G | = n
p n
G
| G | = | Z ( G ) | + [ G : C ( x 1 ) ] + + [ G : C ( x k ) ]
C ( x i )
p
i
i = 1 , , k
C ( x i )
G
p
| C ( x i ) |
C ( x i )
p
G
p
p
p
[ G : C ( x i ) ]
p
G
Z ( G )
Z ( G )
p
G
p
G
G
p
| G | = p n
A 5
| A 5 | = 60 = 2 2 3 5
A 5
2
3
5
A 5
G
p
p r
| G |
G
p r
G
| G | = p
G
n
n > p
n
p
n
| G | = | Z ( G ) | + [ G : C ( x 1 ) ] + + [ G : C ( x k ) ]
p
[ G : C ( x i ) ]
i
p r | C ( x i ) |
p r
| G | = | C ( x i ) | [ G : C ( x i ) ]
C ( x i )
p
[ G : C ( x i ) ]
i
p
| G |
p
| Z ( G ) |
Z ( G )
p
g
N
g
N
Z ( G )
Z ( G )
N
G
Z ( G )
G
G / N
| G | / p
G / N
H
p r 1
H
ϕ : G G / N
p r
G
p
p
p
P
G
p
G
G
S
G
H
S
H
H
S
H × S S
h K h K h 1
K
S
H
N ( H ) = { g G : g H g 1 = H }
G
H
G
H
N ( H )
N ( H )
G
H
P
p
G
x
p
x 1 P x = P
x P
x N ( P )
x P N ( P ) / P
p
H
N ( P )
P
H / P = x P
| H | = | P | | x P |
H
p
P
p
H
P
p
| G |
H = P
H / P
x P = P
x P
H
K
G
H
K
[ H : N ( K ) H ]
K
N ( K ) H
h 1 K h ( N ( K ) H ) h
h 1 , h 2 H
( N ( K ) H ) h 1 = ( N ( K ) H ) h 2
h 2 h 1 1 N ( K )
K = h 2 h 1 1 K h 1 h 2 1
h 1 1 K h 1 = h 2 1 K h 2
H
K
N ( K ) H
H
G
p
| G |
p
G
P 1
P 2
p
g G
g P 1 g 1 = P 2
P
p
G
| G | = p r m
| P | = p r
S = { P = P 1 , P 2 , , P k }
P
G
k = [ G : N ( P ) ]
| G | = p r m = | N ( P ) | [ G : N ( P ) ] = | N ( P ) | k
p r
| N ( P ) |
p
k
p
Q
Q S
Q
P i
S
P i
[ Q : N ( P i ) Q ]
| Q | = [ Q : N ( P i ) Q ] | N ( P i ) Q |
[ Q : N ( P i ) Q ]
| Q | = p r
p
p
k
p
P j
x 1 P j x = P j
x Q
P j = Q
G
p
G
p
1 ( mod p )
| G |
P
p
p
S = { P = P 1 , P 2 , , P k }
P
P
P
p
P
p
| S |
{ P }
| S |
p
1
| S | 1 ( mod p )
G
S
p
P S
| S | = | orbit of P | = [ G : N ( P ) ]
[ G : N ( P ) ]
| G |
p
A 5
2
3
4
5
p
A 5
3
4
5
p
A 5
5
60
1 ( mod 5 )
5
A 5
5
5
A 5
A 5
5
A 5
p
q
p < q
G
p q
q
G
G
q 1 ( mod p )
G
G
H
q
H
p q
1 + k q
k = 0 , 1 ,
1 + q
H
H
G
G
p
K
K
q
1 + k p
k = 0 , 1 ,
q
1 + k p = q
1 + k p = 1
1 + k p = 1
K
G
G
H
K
H
Z q
K
Z p
G Z p × Z q Z p q
15
15 = 5 3
5 1 ( mod 3 )
99 = 3 2 11
G
99
1 + 3 k
3
9
k = 0 , 1 , 2 ,
1 + 3 k
11
3
H
G
1 + 11 k
11
1 + 11 k
9
11
K
G
p 2
p
H
Z 3 × Z 3
Z 9
K
11
Z 11
99
Z 3 × Z 3 × Z 11
Z 9 × Z 11
5 7 47 = 1645
G = a b a 1 b 1 : a , b G
a b a 1 b 1
G
G
G
G / G
G
G
G
5 7 47 = 1645
G
H 1
47
G / H 1
G
H
| G |
1
47
| G | = 1
| G | = 47
G
5
7
H 2
H 3
G
| H 2 | = 5
| H 3 | = 7
G
H i
i = 1 , 2
G
1
5
7
| G | = 1
47
G
G
A 5
G
20
G
5
1 ( mod 5 )
20
1
5
5
G
p n
n > 1
p
G
G
G
4
8
9
16
25
27
32
49
64
81
4
9
25
49
56 = 2 3 7
p
p
7
7
7
8 6 = 48
7
2
2
2
2
48
7
7
2
8
2
56
2
G
G
G
48
48
H
K
G
| H K | = | H | | K | | H K |
H K = { h k : h H , k K }
| H K | | H | | K |
H K
H
K
h 1 k 1 = h 2 k 2
h 1 , h 2 H
k 1 , k 2 K
a = ( h 1 ) 1 h 2 = k 1 ( k 2 ) 1
a H K
( h 1 ) 1 h 2
H
k 2 ( k 1 ) 1
K
h 2 = h 1 a 1 k 2 = a k 1
h = h 1 b 1
k = b k 1
b H K
h k = h 1 k 1
h H
k K
h k H K
h i k i
h i H
k i K
H K
| H K |
| H K | = ( | H | | K | ) / | H K |
G
48
G
8
16
G
2
16
2
H
K
| H K | = 8
| H K | 4
| H K | 16 16 4 = 64
H K
H
K
H
K
H K
H
N ( H K )
N ( H K )
16
| N ( H K ) |
16
1
48
| N ( H K ) | = 48
N ( H K ) = G
p
p
p
69
G
G
p
G
18
24
54
72
80
| G | = 18 = 2 3 2
2
2
3
9
3
S 4
3
S 4
P 1 = { ( 1 ) , ( 123 ) , ( 132 ) }
P 2 = { ( 1 ) , ( 124 ) , ( 142 ) }
P 3 = { ( 1 ) , ( 134 ) , ( 143 ) }
P 4 = { ( 1 ) , ( 234 ) , ( 243 ) }
45
9
H
p
G
H
p
G
N ( H )
96
| G | = 96 = 2 5 3
G
2
2
H
K
| H K | 16
H K
( 32 32 ) / 8 = 128
H K
H
K
2
160
H
G
| H | = p k
p
H
p
G
G
p 2 q 2
p
q
q p 2 1
p q 2 1
G
G
p
p 2
q
q 2
33
3
H
G
H
G
G
p
p
G
G
G
p r
p
G
p r 1
G
p n k
k < p
G
H
G
g N ( H ) g 1 = N ( g H g 1 )
g G
108
175
255
G
G
| G | = 3 5 17
G
p 1 e 1 p n e n
G
n
p
P 1 , , P n
| P i | = p i e i
G
P 1 × × P n
P
p
G
G
P
G
G
| G |
H
G
P
p
H
g G
h
H
g P g 1 = h P h 1
N
P
G = H N
G
p n q
p
q
p > q
G
H
G
[ G : N ( H ) ]
N ( H )
G
H
G
N ( H ) g g 1 H g
2
S 5
D 4
p
p
m
p ( p k m p k )
S
p k
G
p
| S |
G
S
a T = { a t : t T }
a G
T S
p | O T |
T S
{ T 1 , , T u }
p u
H = { g G : g T 1 = T 1 }
H
G
| G | = u | H |
p k
| H |
p k | H |
| H | = | O T | p k
p k = | H |
G
G = a b a 1 b 1 : a , b G
G
G / G
{ a b a 1 b 1 : a , b G }
a G , b G G / G
( a G ) ( b G ) = a b G = a b ( b 1 a 1 b a ) G = ( a b b 1 a 1 ) b a G = b a G
60
G
| G | 60
1
16
14
31
1
46
2
2
17
1
32
51
47
1
3
18
33
1
48
52
4
19
34
49
5
20
5
35
1
50
5
6
21
36
14
51
7
22
2
37
1
52
8
23
1
38
53
9
24
39
2
54
15
10
25
2
40
14
55
2
11
26
2
41
1
56
12
5
27
5
42
57
2
13
28
43
1
58
14
29
1
44
4
59
1
15
1
30
4
45
60
13
G
| G | 60
G
G
n
n = 1 , , 60
n
p
p 0
p
p
p
p
D 18
36 = 2 2 3 2
2
4
3
9
p = 2
2
p = 2
1 , 3
9
2
9
18
2
4
p = 3
1
4
3
3
3
D 18
3
3
6 = 2 3
D 18
D 18
6
64
64
2
26
H S
2
5
44 352 000
40
D 20
p
A 5
A 5
D 36
36
p
p
72
G
36
D 18
H
3
K
6
K
H
H
K
48
4 n
n = 2
p
p
5
A 5
5
A 5
5
S 6
A 5
5
1
p
R
a + b = b + a
a , b R
( a + b ) + c = a + ( b + c )
a , b , c R
0
R
a + 0 = a
a R
a R
a
R
a + ( a ) = 0
( a b ) c = a ( b c )
a , b , c R
a , b , c R
a ( b + c ) = a b + a c ( a + b ) c = a c + b c
( R , + )
1 R
1   0
1 a = a 1 = a
a R
R
R
a b = b a
a , b
R
R
a , b R
a b = 0
a = 0
b = 0
R
R
a R
a   0
a 1
a 1 a = a a 1 = 1
Z
a b = 0
a
b
a = 0
b = 0
Z
2
1 / 2
1
1
Q
R
C
a
b
Z n
a b ( mod n )
Z 12
5 7 11 ( mod 12 )
Z n
Z n
3 4 0 ( mod 12 )
Z 12
a
R
b
R
a b = 0
3
4
Z 12
[ a , b ]
f ( x ) = x 2
g ( x ) = cos x
( f + g ) ( x ) = f ( x ) + g ( x ) = x 2 + cos x
( f g ) ( x ) = f ( x ) g ( x ) = x 2 cos x
2 × 2
R
A B   B A
A B = 0
A
B
1 = ( 1 0 0 1 ) , i = ( 0 1 1 0 ) , j = ( 0 i i 0 ) , k = ( i 0 0 i )
i 2 = 1
i 2 = j 2 = k 2 = 1 i j = k j k = i k i = j j i = k k j = i i k = j
H
a + b i + c j + d k
a , b , c , d
H
2 × 2
( α β β ¯ α ¯ )
α = a + d i
β = b + c i
H
1
i
j
k
( a 1 + b 1 i + c 1 j + d 1 k ) + ( a 2 + b 2 i + c 2 j + d 2 k ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) i + ( c 1 + c 2 ) j + ( d 1 + d 2 ) k
( a 1 + b 1 i + c 1 j + d 1 k ) ( a 2 + b 2 i + c 2 j + d 2 k ) = α + β i + γ j + δ k
α = a 1 a 2 b 1 b 2 c 1 c 2 d 1 d 2 β = a 1 b 2 + a 2 b 1 + c 1 d 2 d 1 c 2 γ = a 1 c 2 b 1 d 2 + c 1 a 2 + d 1 b 2 δ = a 1 d 2 + b 1 c 2 c 1 b 2 + d 1 a 2
H
i
j
k
H
( a + b i + c j + d k ) ( a b i c j d k ) = a 2 + b 2 + c 2 + d 2
a
b
c
d
a + b i + c j + d k   0
( a + b i + c j + d k ) ( a b i c j d k a 2 + b 2 + c 2 + d 2 ) = 1
R
a , b R
a 0 = 0 a = 0
a ( b ) = ( a ) b = a b
( a ) ( b ) = a b
a 0 = a ( 0 + 0 ) = a 0 + a 0 ;
a 0 = 0
0 a = 0
a b + a ( b ) = a ( b b ) = a 0 = 0
a b = a ( b )
a b = ( a ) b
( a ) ( b ) = ( a ( b ) ) = ( a b ) = a b
S
R
S
R
S
R
n Z
Z
Z Q R C
R
S
R
S
R
S  
r s S
r , s S
r s S
r , s S
R = M 2 ( R )
2 × 2
R
T
R
T = { ( a b 0 c ) : a , b , c R }
T
R
A = ( a b 0 c ) and B = ( a b 0 c )
T
A B
T
A B = ( a a a b + b c 0 c c )
T
R
r
R
r
s R
r s = 0
a
R
a
R
R
i 2 = 1
Z [ i ] = { m + n i : m , n Z }
α = a + b i
Z [ i ]
α ¯ = a b i
α β = 1
α ¯ β ¯ = 1
β = c + d i
1 = α β α ¯ β ¯ = ( a 2 + b 2 ) ( c 2 + d 2 )
a 2 + b 2
1
1
a + b i = ± 1
a + b i = ± i
± 1
± i
F = { ( 1 0 0 1 ) , ( 1 1 1 0 ) , ( 0 1 1 1 ) , ( 0 0 0 0 ) }
Z 2
Q ( 2 ) = { a + b 2 : a , b Q }
a + b 2
Q ( 2 )
a a 2 2 b 2 + b a 2 2 b 2 2
D
D
a D
a b = a c
b = c
D
D
a b = a c
a   0
a ( b c ) = 0
b c = 0
b = c
D
a b = a c
b = c
a b = 0
a   0
a b = a 0
b = 0
a
D
D
D
D
a D
λ a : D D
λ a ( d ) = a d
a   0
d   0
a d   0
λ a
d 1 , d 2 D
a d 1 = λ a ( d 1 ) = λ a ( d 2 ) = a d 2
d 1 = d 2
D
λ a
d D
λ a ( d ) = a d = 1
a
D
d
a
D
n
r
R
r + + r
n
n r
R
n
n r = 0
r R
R
0
R
char R
R
p
Z p
p
Z p
Z p
a
p a = 0
Z p
p
R
1
n
R
n
1
n
n
n 1 = 0
r R
n r = n ( 1 r ) = ( n 1 ) r = 0 r = 0
n
n 1 = 0
R
D
D
n
n   0
n
n = a b
1 < a < n
1 < b < n
n 1 = 0
0 = n 1 = ( a b ) 1 = ( a 1 ) ( b 1 )
D
a 1 = 0
b 1 = 0
D
n
n
R
S
ϕ : R S
ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) ϕ ( a b ) = ϕ ( a ) ϕ ( b )
a , b R
ϕ : R S
ϕ
0
ϕ : R S
ker ϕ = { r R : ϕ ( r ) = 0 }
n
ϕ : Z Z n
a a ( mod n )
ϕ ( a + b ) = ( a + b ) ( mod n ) = a ( mod n ) + b ( mod n ) = ϕ ( a ) + ϕ ( b )
ϕ ( a b ) = a b ( mod n ) = a ( mod n ) b ( mod n ) = ϕ ( a ) ϕ ( b )
ϕ
n Z
C [ a , b ]
[ a , b ]
α [ a , b ]
ϕ α : C [ a , b ] R
ϕ α ( f ) = f ( α )
ϕ α ( f + g ) = ( f + g ) ( α ) = f ( α ) + g ( α ) = ϕ α ( f ) + ϕ α ( g ) ϕ α ( f g ) = ( f g ) ( α ) = f ( α ) g ( α ) = ϕ α ( f ) ϕ α ( g )
ϕ α
ϕ : R S
R
ϕ ( R )
ϕ ( 0 ) = 0
1 R
1 S
R
S
ϕ
ϕ ( 1 R ) = 1 S
R
ϕ ( R )   { 0 }
ϕ ( R )
R
I
R
a
I
r
R
a r
r a
I
r I I
I r I
r R
R
{ 0 }
R
R
I
R
1
I
r R
r 1 = r I
I = R
a
R
a = { a r : r R }
R
a
0 = a 0
a = a 1
a
a
a
a r + a r = a ( r + r )
a r
a r = a ( r ) a
a r a
s R
s ( a r ) = a ( s r )
a
R
a = { a r : r R }
Z
{ 0 }
0 = { 0 }
I
Z
I
m
n
I
a
I
q
r
a = n q + r
0 r < n
r = a n q I
r
0
n
I
a = n q
I = n
n Z
n a
n Z
b
Z
n a b
n Z
Z
ϕ : R S
R
ker ϕ
R
r R
a ker ϕ
a r
r a
ker ϕ
ϕ ( a r ) = ϕ ( a ) ϕ ( r ) = 0 ϕ ( r ) = 0
ϕ ( r a ) = ϕ ( r ) ϕ ( a ) = ϕ ( r ) 0 = 0
r I I
I r I
r R
r I I
I r I
r R
I
R
R / I
( r + I ) ( s + I ) = r s + I
R / I
r + I
s + I
R / I
( r + I ) ( s + I ) = r s + I
r r + I
s s + I
r s
r s + I
r r + I
a
I
r = r + a
b I
s = s + b
r s = ( r + a ) ( s + b ) = r s + a s + r b + a b
a s + r b + a b I
I
r s r s + I
R / I
I
R
ϕ : R R / I
ϕ ( r ) = r + I
R
R / I
I
ϕ : R R / I
ϕ
r
s
R
ϕ ( r ) ϕ ( s ) = ( r + I ) ( s + I ) = r s + I = ϕ ( r s )
ϕ : R R / I
ψ : R S
ker ψ
R
ϕ : R R / ker ψ
η : R / ker ψ ψ ( R )
ψ = η ϕ
K = ker ψ
η : R / K ψ ( R )
η ( r + K ) = ψ ( r )
R
R / K
η ( ( r + K ) ( s + K ) ) = η ( r + K ) η ( s + K )
η ( ( r + K ) ( s + K ) ) = η ( r s + K ) = ψ ( r s ) = ψ ( r ) ψ ( s ) = η ( r + K ) η ( s + K )
I
R
J
R
I J
I
I / I J ( I + J ) / J
R
I
J
R
J I
R / I R / J I / J
I
R
S S / I
S
I
R / I
R
I
R / I
I
R
R / I
M
R
R
M
R
R
M
I
M
I = R
R
M
R
M
R
R / M
M
R
R
R / M
1 + M
R / M
R / M
a + M
R / M
a M
I
{ r a + m : r R and m M }
I
R
I
0 a + 0 = 0
I
r 1 a + m 1
r 2 a + m 2
I
( r 1 a + m 1 ) ( r 2 a + m 2 ) = ( r 1 r 2 ) a + ( m 1 m 2 )
I
r R
r I I
I
I
M
M
I = R
I
m
M
b
R
1 = a b + m
1 + M = a b + M = b a + M = ( a + M ) ( b + M )
M
R / M
R / M
0 + M = M
1 + M
M
R
I
M
I = R
a
I
M
a + M
b + M
R / M
( a + M ) ( b + M ) = a b + M = 1 + M
m M
a b + m = 1
1
I
r 1 = r I
r R
I = R
p Z
Z
p
p Z
Z / p Z Z p
P
R
a b P
a P
b P
P = { 0 , 2 , 4 , 6 , 8 , 10 }
Z 12
R
1
1   0
P
R
R / P
P
R
R / P
a b P
a + P
b + P
R / P
( a + P ) ( b + P ) = 0 + P = P
a + P = P
b + P = P
a
P
b
P
P
P
( a + P ) ( b + P ) = a b + P = 0 + P = P
a b P
a P
b
P
b + P = 0 + P
R / P
Z
n Z
Z / n Z Z n
n
Z
p Z
p
m
n
gcd ( m , n ) = 1
a , b Z
x a ( mod m ) x b ( mod n )
x 1
x 2
x 1 x 2 ( mod m n )
x a ( mod m )
a + k m
k Z
k 1
a + k 1 m b ( mod n )
k 1 m ( b a ) ( mod n )
k 1
m
n
s
t
m s + n t = 1
( b a ) m s = ( b a ) ( b a ) n t
[ ( b a ) s ] m ( b a ) ( mod n )
k 1 = ( b a ) s
m n
c 1
c 2
c i a ( mod m ) c i b ( mod n )
i = 1 , 2
c 2 c 1 ( mod m ) c 2 c 1 ( mod n )
m
n
c 1 c 2
c 2 c 1 ( mod m n )
x 3 ( mod 4 ) x 4 ( mod 5 )
s
t
4 s + 5 t = 1
s = 4
t = 3
x = a + k 1 m = 3 + 4 k 1 = 3 + 4 [ ( 5 4 ) 4 ] = 19
n 1 , n 2 , , n k
gcd ( n i , n j ) = 1
i   j
a 1 , , a k
x a 1 ( mod n 1 ) x a 2 ( mod n 2 ) x a k ( mod n k )
n 1 n 2 n k
k = 2
k
x a 1 ( mod n 1 ) x a 2 ( mod n 2 ) x a k + 1 ( mod n k + 1 )
k
n 1 n k
a
n 1 n k
n k + 1
x a ( mod n 1 n k ) x a k + 1 ( mod n k + 1 )
n 1 n k + 1
x 3 ( mod 4 ) x 4 ( mod 5 ) x 1 ( mod 9 ) x 5 ( mod 7 )
19
19 ( mod 20 )
x 19 ( mod 20 ) x 1 ( mod 9 ) x 5 ( mod 7 )
x 19 ( mod 180 ) x 5 ( mod 7 )
19
1260
2 63 1 = 9,223,372,036,854,775,807
2 511 1
2134
1531
95
97
98
99
2134 44 ( mod 95 ) 2134 0 ( mod 97 ) 2134 76 ( mod 98 ) 2134 55 ( mod 99 )
1531 11 ( mod 95 ) 1531 76 ( mod 97 ) 1531 61 ( mod 98 ) 1531 46 ( mod 99 )
2134 1531 44 11 9 ( mod 95 ) 2134 1531 0 76 0 ( mod 97 ) 2134 1531 76 61 30 ( mod 98 ) 2134 1531 55 46 55 ( mod 99 )
2134 1531
x 9 ( mod 95 ) x 0 ( mod 97 ) x 30 ( mod 98 ) x 55 ( mod 99 )
95 97 98 99 = 89,403,930
x
2134 1531 = 3,267,154
7 Z
Z 18
Q ( 2 ) = { a + b 2 : a , b Q }
Q ( 2 , 3 ) = { a + b 2 + c 3 + d 6 : a , b , c , d Q }
Z [ 3 ] = { a + b 3 : a , b Z }
R = { a + b 3 3 : a , b Q }
Z [ i ] = { a + b i : a , b Z and i 2 = 1 }
Q ( 3 3 ) = { a + b 3 3 + c 9 3 : a , b , c Q }
7 Z
Q ( 2 )
R
R
2 × 2
( a b 0 0 )
a , b R
R
S
R
Z 10
Z 12
Z 7
M 2 ( Z )
2 × 2
Z
M 2 ( Z 2 )
2 × 2
Z 2
{ 1 , 3 , 7 , 9 }
{ 1 , 2 , 3 , 4 , 5 , 6 }
{ ( 1 0 0 1 ) , ( 1 1 0 1 ) , ( 1 0 1 1 ) , ( 0 1 1 0 ) , ( 1 1 1 0 ) , ( 0 1 1 1 ) , }
Z 18
Z 25
M 2 ( R )
2 × 2
R
M 2 ( Z )
2 × 2
Z
Q
{ 0 }
{ 0 , 9 }
{ 0 , 6 , 12 }
{ 0 , 3 , 6 , 9 , 12 , 15 }
{ 0 , 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 }
R
I
R / I
R = Z
I = 6 Z
R = Z 12
I = { 0 , 3 , 6 , 9 }
ϕ : Z / 6 Z Z / 15 Z
R
C
ϕ : C R
ϕ ( i ) = a
Q ( 2 ) = { a + b 2 : a , b Q }
Q ( 3 ) = { a + b 3 : a , b Q }
ϕ : Q ( 2 ) Q ( 3 )
ϕ ( 2 ) = a
F = { ( 1 0 0 1 ) , ( 1 1 1 0 ) , ( 0 1 1 1 ) , ( 0 0 0 0 ) }
Z 2
ϕ : C M 2 ( R )
ϕ ( a + b i ) = ( a b b a )
ϕ
C
M 2 ( R )
Z [ i ]
Z [ 3 i ] = { a + b 3 i : a , b Z }
x 2 ( mod 5 ) x 6 ( mod 11 )
x 3 ( mod 7 ) x 0 ( mod 8 ) x 5 ( mod 15 )
x 2 ( mod 4 ) x 4 ( mod 7 ) x 7 ( mod 9 ) x 5 ( mod 11 )
x 3 ( mod 5 ) x 0 ( mod 8 ) x 1 ( mod 11 ) x 5 ( mod 13 )
x 17 ( mod 55 )
x 214 ( mod 2772 )
2234 + 4121
95
97
98
99
2134 1531
98
99
R
R
{ 0 }
R
I   { 0 }
1 I
a
R
( 1 ) a = a
ϕ : R S
R
ϕ ( R )
ϕ ( 0 ) = 0
1 R
1 S
R
S
ϕ
ϕ ( 1 R ) = 1 S
R
ϕ ( R )   0
ϕ ( R )
ϕ ( a ) ϕ ( b ) = ϕ ( a b ) = ϕ ( b a ) = ϕ ( b ) ϕ ( a )
R / I
I
R
J
R
I J
I
I / I J I + J / J
R
I
J
R
J I
R / I R / J I / J
I
R
S S / I
S
I
R / I
R
R / I
R
S
R
S
R
S  
r s S
r , s S
r s S
r , s S
R
{ R α }
R α
R
{ I α } α A
R
α A I α
R
I 1
I 2
R
I 1 I 2
R
R
{ 0 }
R
R
a R
a   0
a
R
b R
a b = 1
R
a
R
a n = 0
n
R
R
a R
a 2 = a
( a + b ) 2
( a b ) 2
R
a 3 = a
a R
R
R
1 R
S
R
1 S
1 R = 1 S
R
1 = 0
R = { 0 }
R
R
Z ( R ) = { a R : a r = r a for all r R }
Z ( R )
R
p
Z ( p ) = { a / b : a , b Z and gcd ( b , p ) = 1 }
Z ( p )
p
p
a / b , c / d Z ( p )
a / b + c / d = ( a d + b c ) / b d
( a / b ) ( c / d ) = ( a c ) / ( b d )
Z ( p )
gcd ( b d , p ) = 1
Z p
R
u
R
i u : R R
r u r u 1
i u
R
R
R
R
Inn ( R )
R
Aut ( R )
Inn ( R )
Aut ( R )
U ( R )
R
ϕ : U ( R ) Inn ( R )
u i u
ϕ
Aut ( Z )
Inn ( Z )
U ( Z )
R
S
R × S
( r , s ) + ( r , s ) = ( r + r , s + s )
( r , s ) ( r , s ) = ( r r , s s )
x
x 2 = x
0
1
x
x 2 = x
x   0
R
x = 1
M 2 ( R )
gcd ( a , n ) = d
gcd ( b , d )   1
a x b ( mod n )
R
I
J
R
I + J = R
r
s
R
x r ( mod I ) x s ( mod J )
I J
I
J
R
I + J = R
R / ( I J ) R / I × R / J
Z
n
Z n
n
n
Q
R
C
x 2 n = 0
Q [ n ]
x n 1 = 0
Q ¯
p
Z p
7
x 2 7
7
7
x 2 7
x 2 7
r 2 = n
s 2 = m
t = r s = s r
Z
Z 4
y
y ¯
Z 11
Z 12
1
Z
4
{ a 3 + b 5 a , b Z }
Z
F
F
F
F
Z 3
P
z
Z 7
K
z 2 + z + 3
H
P
K
H
p ( x ) = x 3 3 x + 2 q ( x ) = 3 x 2 6 x + 5
p ( x ) + q ( x )
p ( x ) q ( x )
( p + q ) ( x ) = p ( x ) + q ( x ) = ( x 3 3 x + 2 ) + ( 3 x 2 6 x + 5 ) = x 3 + 3 x 2 9 x + 7
( p q ) ( x ) = p ( x ) q ( x ) = ( x 3 3 x + 2 ) ( 3 x 2 6 x + 5 ) = 3 x 5 6 x 4 4 x 3 + 24 x 2 27 x + 10
R
f ( x ) = i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n
a i R
a n   0
R
x
a 0 , a 1 , , a n
f
a n
n
a n   0
f
n
deg f ( x ) = n
n
f = 0
f
R
R [ x ]
R
p ( x ) = a 0 + a 1 x + + a n x n q ( x ) = b 0 + b 1 x + + b m x m
p ( x ) = q ( x )
a i = b i
i 0
p ( x ) = a 0 + a 1 x + + a n x n q ( x ) = b 0 + b 1 x + + b m x m
p ( x )
q ( x )
p ( x ) + q ( x ) = c 0 + c 1 x + + c k x k
c i = a i + b i
i
p ( x )
q ( x )
p ( x ) q ( x ) = c 0 + c 1 x + + c m + n x m + n
c i = k = 0 i a k b i k = a 0 b i + a 1 b i 1 + + a i 1 b 1 + a i b 0
i
p ( x ) = 3 + 0 x + 0 x 2 + 2 x 3 + 0 x 4
q ( x ) = 2 + 0 x x 2 + 0 x 3 + 4 x 4
Z [ x ]
p ( x ) = 3 + 2 x 3
q ( x ) = 2 x 2 + 4 x 4
p ( x ) + q ( x ) = 5 x 2 + 2 x 3 + 4 x 4
p ( x ) q ( x ) = ( 3 + 2 x 3 ) ( 2 x 2 + 4 x 4 ) = 6 3 x 2 + 4 x 3 + 12 x 4 2 x 5 + 8 x 7
c i
p ( x ) = 3 + 3 x 3 and q ( x ) = 4 + 4 x 2 + 4 x 4
Z 12 [ x ]
p ( x )
q ( x )
7 + 4 x 2 + 3 x 3 + 4 x 4
R [ x ]
R
R
R [ x ]
R [ x ]
f ( x ) = 0
p ( x ) = i = 0 n a i x i
p ( x )
p ( x ) = i = 0 n ( a i ) x i = i = 0 n a i x i
R
p ( x ) = i = 0 m a i x i , q ( x ) = i = 0 n b i x i , r ( x ) = i = 0 p c i x i
[ p ( x ) q ( x ) ] r ( x ) = [ ( i = 0 m a i x i ) ( i = 0 n b i x i ) ] ( i = 0 p c i x i ) = [ i = 0 m + n ( j = 0 i a j b i j ) x i ] ( i = 0 p c i x i ) = i = 0 m + n + p [ j = 0 i ( k = 0 j a k b j k ) c i j ] x i = i = 0 m + n + p ( j + k + l = i a j b k c l ) x i = i = 0 m + n + p [ j = 0 i a j ( k = 0 i j b k c i j k ) ] x i = ( i = 0 m a i x i ) [ i = 0 n + p ( j = 0 i b j c i j ) x i ] = ( i = 0 m a i x i ) [ ( i = 0 n b i x i ) ( i = 0 p c i x i ) ] = p ( x ) [ q ( x ) r ( x ) ]
p ( x )
q ( x )
R [ x ]
R
deg p ( x ) + deg q ( x ) = deg ( p ( x ) q ( x ) )
R [ x ]
p ( x ) = a m x m + + a 1 x + a 0
q ( x ) = b n x n + + b 1 x + b 0
a m   0
b n   0
p ( x )
q ( x )
m
n
p ( x ) q ( x )
a m b n x m + n
R
p ( x ) q ( x )
m + n
p ( x ) q ( x )   0
p ( x )   0
q ( x )   0
p ( x ) q ( x )   0
R [ x ]
x 2 3 x y + 2 y 3
R
x
y
( R [ x ] ) [ y ]
( R [ x ] ) [ y ] R ( [ y ] ) [ x ]
R [ x , y ]
R [ x , y ]
x
y
R
n
n
R
R [ x 1 , x 2 , , x n ]
n
R
α R
ϕ α : R [ x ] R
α
ϕ α ( p ( x ) ) = p ( α ) = a n α n + + a 1 α + a 0
p ( x ) = a n x n + + a 1 x + a 0
p ( x ) = i = 0 n a i x i
q ( x ) = i = 0 m b i x i
ϕ α ( p ( x ) + q ( x ) ) = ϕ α ( p ( x ) ) + ϕ α ( q ( x ) )
ϕ α
ϕ α ( p ( x ) ) ϕ α ( q ( x ) ) = p ( α ) q ( α ) = ( i = 0 n a i α i ) ( i = 0 m b i α i ) = i = 0 m + n ( k = 0 i a k b i k ) α i = ϕ α ( p ( x ) q ( x ) )
ϕ α : R [ x ] R
α
a
b
b > 0
q
r
a = b q + r
0 r < b
q
r
f ( x )
g ( x )
F [ x ]
F
g ( x )
q ( x ) , r ( x ) F [ x ]
f ( x ) = g ( x ) q ( x ) + r ( x )
deg r ( x ) < deg g ( x )
r ( x )
q ( x )
r ( x )
f ( x )
0 = 0 g ( x ) + 0 ;
q
r
f ( x )
deg f ( x ) = n
deg g ( x ) = m
m > n
q ( x ) = 0
r ( x ) = f ( x )
m n
n
f ( x ) = a n x n + a n 1 x n 1 + + a 1 x + a 0 g ( x ) = b m x m + b m 1 x m 1 + + b 1 x + b 0
f ( x ) = f ( x ) a n b m x n m g ( x )
n
q ( x )
r ( x )
f ( x ) = q ( x ) g ( x ) + r ( x )
r ( x ) = 0
r ( x )
g ( x )
q ( x ) = q ( x ) + a n b m x n m
f ( x ) = g ( x ) q ( x ) + r ( x )
r ( x )
deg r ( x ) < deg g ( x )
q ( x )
r ( x )
q 1 ( x )
r 1 ( x )
f ( x ) = g ( x ) q 1 ( x ) + r 1 ( x )
deg r 1 ( x ) < deg g ( x )
r 1 ( x ) = 0
f ( x ) = g ( x ) q ( x ) + r ( x ) = g ( x ) q 1 ( x ) + r 1 ( x )
g ( x ) [ q ( x ) q 1 ( x ) ] = r 1 ( x ) r ( x )
q ( x ) q 1 ( x )
deg ( g ( x ) [ q ( x ) q 1 ( x ) ] ) = deg ( r 1 ( x ) r ( x ) ) deg g ( x )
r ( x )
r 1 ( x )
g ( x )
r ( x ) = r 1 ( x )
q ( x ) = q 1 ( x )
x 3 x 2 + 2 x 3
x 2
x 2
+
x
+
4
x
2
x 3
x 2
+
2 x
3
x 3
2 x 2
x 2
+
2 x
3
x 2
2 x
4 x
3
4 x
8
5
x 3 x 2 + 2 x 3 = ( x 2 ) ( x 2 + x + 4 ) + 5
p ( x )
F [ x ]
α F
α
p ( x )
p ( x )
ϕ α
α
p ( x )
p ( α ) = 0
F
α F
p ( x ) F [ x ]
x α
p ( x )
F [ x ]
α F
p ( α ) = 0
q ( x )
r ( x )
p ( x ) = ( x α ) q ( x ) + r ( x )
r ( x )
x α
r ( x )
r ( x ) = a
a F
p ( x ) = ( x α ) q ( x ) + a
0 = p ( α ) = 0 q ( α ) + a = a ;
p ( x ) = ( x α ) q ( x )
x α
p ( x )
x α
p ( x )
p ( x ) = ( x α ) q ( x )
p ( α ) = 0 q ( α ) = 0
F
p ( x )
n
F [ x ]
n
F
p ( x )
deg p ( x ) = 0
p ( x )
deg p ( x ) = 1
p ( x ) = a x + b
a
b
F
α 1
α 2
p ( x )
a α 1 + b = a α 2 + b
α 1 = α 2
deg p ( x ) > 1
p ( x )
F
α
p ( x )
p ( x ) = ( x α ) q ( x )
q ( x ) F [ x ]
q ( x )
n 1
β
p ( x )
α
p ( β ) = ( β α ) q ( β ) = 0
α   β
F
q ( β ) = 0
q ( x )
n 1
F
α
p ( x )
n
F
F
d ( x )
p ( x ) , q ( x ) F [ x ]
d ( x )
p ( x )
q ( x )
d ( x )
p ( x )
q ( x )
d ( x ) d ( x )
d ( x ) = gcd ( p ( x ) , q ( x ) )
p ( x )
q ( x )
gcd ( p ( x ) , q ( x ) ) = 1
F
d ( x )
p ( x )
q ( x )
F [ x ]
r ( x )
s ( x )
d ( x ) = r ( x ) p ( x ) + s ( x ) q ( x )
d ( x )
S = { f ( x ) p ( x ) + g ( x ) q ( x ) : f ( x ) , g ( x ) F [ x ] }
d ( x ) = r ( x ) p ( x ) + s ( x ) q ( x )
r ( x )
s ( x )
F [ x ]
d ( x )
p ( x )
q ( x )
d ( x )
p ( x )
a ( x )
b ( x )
p ( x ) = a ( x ) d ( x ) + b ( x )
b ( x )
deg b ( x ) < deg d ( x )
b ( x ) = p ( x ) a ( x ) d ( x ) = p ( x ) a ( x ) ( r ( x ) p ( x ) + s ( x ) q ( x ) ) = p ( x ) a ( x ) r ( x ) p ( x ) a ( x ) s ( x ) q ( x ) = p ( x ) ( 1 a ( x ) r ( x ) ) + q ( x ) ( a ( x ) s ( x ) )
p ( x )
q ( x )
S
b ( x )
d ( x )
d ( x )
p ( x )
d ( x )
q ( x )
d ( x )
p ( x )
q ( x )
d ( x )
p ( x )
q ( x )
d ( x )
p ( x )
q ( x )
d ( x ) d ( x )
d ( x )
p ( x )
q ( x )
u ( x )
v ( x )
p ( x ) = u ( x ) d ( x )
q ( x ) = v ( x ) d ( x )
d ( x ) = r ( x ) p ( x ) + s ( x ) q ( x ) = r ( x ) u ( x ) d ( x ) + s ( x ) v ( x ) d ( x ) = d ( x ) [ r ( x ) u ( x ) + s ( x ) v ( x ) ]
d ( x ) d ( x )
d ( x )
p ( x )
q ( x )
p ( x )
q ( x )
d ( x )
p ( x )
q ( x )
u ( x )
v ( x )
F [ x ]
d ( x ) = d ( x ) [ r ( x ) u ( x ) + s ( x ) v ( x ) ]
deg d ( x ) = deg d ( x ) + deg [ r ( x ) u ( x ) + s ( x ) v ( x ) ]
d ( x )
d ( x )
deg d ( x ) = deg d ( x )
d ( x )
d ( x )
d ( x ) = d ( x )
f ( x ) F [ x ]
F
f ( x )
g ( x )
h ( x )
F [ x ]
g ( x )
h ( x )
f ( x )
x 2 2 Q [ x ]
x 2 + 1
p ( x ) = x 3 + x 2 + 2
Z 3 [ x ]
Z 3 [ x ]
x a
a
Z 3 [ x ]
p ( a ) = 0
p ( 0 ) = 2 p ( 1 ) = 1 p ( 2 ) = 2
p ( x )
Z 3
p ( x ) Q [ x ]
p ( x ) = r s ( a 0 + a 1 x + + a n x n )
r , s , a 0 , , a n
a i
r
s
p ( x ) = b 0 c 0 + b 1 c 1 x + + b n c n x n
b i
c i
p ( x )
p ( x ) = 1 c 0 c n ( d 0 + d 1 x + + d n x n )
d 0 , , d n
d
d 0 , , d n
p ( x ) = d c 0 c n ( a 0 + a 1 x + + a n x n )
d i = d a i
a i
d / ( c 0 c n )
p ( x ) = r s ( a 0 + a 1 x + + a n x n )
gcd ( r , s ) = 1
p ( x ) Z [ x ]
p ( x )
α ( x )
β ( x )
Q [ x ]
α ( x )
β ( x )
p ( x )
p ( x ) = a ( x ) b ( x )
a ( x )
b ( x )
Z [ x ]
deg α ( x ) = deg a ( x )
deg β ( x ) = deg b ( x )
α ( x ) = c 1 d 1 ( a 0 + a 1 x + + a m x m ) = c 1 d 1 α 1 ( x ) β ( x ) = c 2 d 2 ( b 0 + b 1 x + + b n x n ) = c 2 d 2 β 1 ( x )
a i
b i
p ( x ) = α ( x ) β ( x ) = c 1 c 2 d 1 d 2 α 1 ( x ) β 1 ( x ) = c d α 1 ( x ) β 1 ( x )
c / d
c 1 / d 1
c 2 / d 2
d p ( x ) = c α 1 ( x ) β 1 ( x )
d = 1
c a m b n = 1
p ( x )
c = 1
c = 1
c = 1
a m = b n = 1
a m = b n = 1
p ( x ) = α 1 ( x ) β 1 ( x )
α 1 ( x )
β 1 ( x )
deg α ( x ) = deg α 1 ( x )
deg β ( x ) = deg β 1 ( x )
a ( x ) = α 1 ( x )
b ( x ) = β 1 ( x )
p ( x ) = ( α 1 ( x ) ) ( β 1 ( x ) ) = a ( x ) b ( x )
c = 1
d   1
gcd ( c , d ) = 1
p
p d
p c
α 1 ( x )
a i
p a i
b j
β 1 ( x )
p b j
α 1 ( x )
β 1 ( x )
Z p [ x ]
α 1 ( x )
β 1 ( x )
p
p d
α 1 ( x ) β 1 ( x ) = 0
Z p [ x ]
α 1 ( x )
β 1 ( x )
Z p [ x ]
d = 1
p ( x ) = x n + a n 1 x n 1 + + a 0
Z
a 0   0
p ( x )
Q
p ( x )
α
Z
α
a 0
p ( x )
a Q
p ( x )
x a
p ( x )
Z [ x ]
α Z
p ( x ) = ( x α ) ( x n 1 + a 0 / α )
a 0 / α Z
α a 0
p ( x ) = x 4 2 x 3 + x + 1
p ( x )
Q [ x ]
p ( x )
p ( x )
p ( x ) = ( x α ) q ( x )
q ( x )
p ( x )
p ( x )
Q [ x ]
Z
± 1
p ( 1 ) = 1
p ( 1 ) = 3
p ( x )
p ( x )
p ( x ) = ( x 2 + a x + b ) ( x 2 + c x + d ) = x 4 + ( a + c ) x 3 + ( a c + b + d ) x 2 + ( a d + b c ) x + b d
Z [ x ]
a + c = 2 a c + b + d = 0 a d + b c = 1 b d = 1
b d = 1
b = d = 1
b = d = 1
b = d
a d + b c = b ( a + c ) = 1
a + c = 2
2 b = 1
b
p ( x )
Q
p
f ( x ) = a n x n + + a 0 Z [ x ]
p a i
i = 0 , 1 , , n 1
p a n
p 2 a 0
f ( x )
Q
f ( x )
Z [ x ]
f ( x ) = ( b r x r + + b 0 ) ( c s x s + + c 0 )
Z [ x ]
b r
c s
r , s < n
p 2
a 0 = b 0 c 0
b 0
c 0
p
p b 0
p c 0
p a n
a n = b r c s
b r
c s
p
m
k
p c k
a m = b 0 c m + b 1 c m 1 + + b m c 0
p
p
b 0 c m
m = n
a i
p
m < n
f ( x )
f ( x ) = 16 x 5 9 x 4 + 3 x 2 + 6 x 21
Q
p = 3
Q
Q [ x ]
F [ x ]
F
F [ x ]
p ( x )
p ( x )
p ( x ) = { p ( x ) q ( x ) : q ( x ) F [ x ] }
x 2
F [ x ]
x 2
1
F
F [ x ]
I
F [ x ]
I
I
F [ x ]
p ( x ) I
deg p ( x ) = 0
p ( x )
I
F [ x ]
1 = I = F [ x ]
I
deg p ( x ) 1
f ( x )
I
q ( x )
r ( x )
F [ x ]
f ( x ) = p ( x ) q ( x ) + r ( x )
deg r ( x ) < deg p ( x )
f ( x ) , p ( x ) I
I
r ( x ) = f ( x ) p ( x ) q ( x )
I
p ( x )
r ( x )
f ( x )
I
p ( x ) q ( x )
q ( x ) F [ x ]
I = p ( x )
F [ x , y ]
F [ x , y ]
x
y
F [ x , y ]
x
y
F
p ( x ) F [ x ]
p ( x )
p ( x )
p ( x )
F [ x ]
p ( x )
F [ x ]
F [ x ]
p ( x )
p ( x )
p ( x ) = f ( x ) g ( x )
p ( x )
f ( x )
p ( x )
p ( x )
p ( x ) f ( x )
p ( x )
p ( x )
F [ x ]
I
F [ x ]
p ( x )
I
I = f ( x )
f ( x ) F [ x ]
p ( x ) I
p ( x ) = f ( x ) g ( x )
g ( x ) F [ x ]
p ( x )
f ( x )
g ( x )
f ( x )
I = F [ x ]
g ( x )
f ( x )
I
I = p ( x )
F [ x ]
p ( x )
a x 2 + b x + c = 0
a x 3 + b x 2 + c x + d = 0
a x 3 + c x + d = 0
a x 3 + b x 2 + c x + d = 0
a x 4 + b x 3 + c x 2 + d x + e = 0
p ( x )
n
p ( x )
8 x 5 18 x 4 + 20 x 3 25 x 2 + 20
4 x 2 x 2
3
Z 2 [ x ]
( 5 x 2 + 3 x 4 ) + ( 4 x 2 x + 9 )
Z 12
( 5 x 2 + 3 x 4 ) ( 4 x 2 x + 9 )
Z 12
( 7 x 3 + 3 x 2 x ) + ( 6 x 2 8 x + 4 )
Z 9
( 3 x 2 + 2 x 4 ) + ( 4 x 2 + 2 )
Z 5
( 3 x 2 + 2 x 4 ) ( 4 x 2 + 2 )
Z 5
( 5 x 2 + 3 x 2 ) 2
Z 12
9 x 2 + 2 x + 5
8 x 4 + 7 x 3 + 2 x 2 + 7 x
q ( x )
r ( x )
a ( x ) = q ( x ) b ( x ) + r ( x )
deg r ( x ) < deg b ( x )
a ( x ) = 5 x 3 + 6 x 2 3 x + 4
b ( x ) = x 2
Z 7 [ x ]
a ( x ) = 6 x 4 2 x 3 + x 2 3 x + 1
b ( x ) = x 2 + x 2
Z 7 [ x ]
a ( x ) = 4 x 5 x 3 + x 2 + 4
b ( x ) = x 3 2
Z 5 [ x ]
a ( x ) = x 5 + x 3 x 2 x
b ( x ) = x 3 + x
Z 2 [ x ]
5 x 3 + 6 x 2 3 x + 4 = ( 5 x 2 + 2 x + 1 ) ( x 2 ) + 6
4 x 5 x 3 + x 2 + 4 = ( 4 x 2 + 4 ) ( x 3 + 3 ) + 4 x 2 + 2
p ( x )
q ( x )
d ( x ) = gcd ( p ( x ) , q ( x ) )
a ( x )
b ( x )
a ( x ) p ( x ) + b ( x ) q ( x ) = d ( x )
p ( x ) = x 3 6 x 2 + 14 x 15
q ( x ) = x 3 8 x 2 + 21 x 18
p ( x ) , q ( x ) Q [ x ]
p ( x ) = x 3 + x 2 x + 1
q ( x ) = x 3 + x 1
p ( x ) , q ( x ) Z 2 [ x ]
p ( x ) = x 3 + x 2 4 x + 4
q ( x ) = x 3 + 3 x 2
p ( x ) , q ( x ) Z 5 [ x ]
p ( x ) = x 3 2 x + 4
q ( x ) = 4 x 3 + x + 3
p ( x ) , q ( x ) Q [ x ]
5 x 3 + 4 x 2 x + 9
Z 12
3 x 3 4 x 2 x + 4
Z 5
5 x 4 + 2 x 2 3
Z 7
x 3 + x + 1
Z 2
Z 12
3
4
Z [ x ]
p ( x )
Z 4 [ x ]
deg p ( x ) > 1
( 2 x + 1 )
Q [ x ]
x 4 2 x 3 + 2 x 2 + x + 4
x 4 5 x 3 + 3 x 2
3 x 5 4 x 3 6 x 2 + 6
5 x 5 6 x 4 3 x 2 + 9 x 15
2
3
Z 2 [ x ]
x 2 + x + 8
Z 10 [ x ]
x 2 + x + 8 = ( x + 2 ) ( x + 9 )
p ( x )
Z 6 [ x ]
n
n
F
F [ x 1 , , x n ]
Z [ x ]
Z
x p + a
a Z p
p
f ( x )
F [ x ]
F
f ( x ) p ( x ) q ( x )
f ( x ) p ( x )
f ( x ) q ( x )
R
S
R [ x ] S [ x ]
ϕ : R S
ϕ ¯ : R [ x ] S [ x ]
ϕ ¯ ( a 0 + a 1 x + + a n x n ) = ϕ ( a 0 ) + ϕ ( a 1 ) x + + ϕ ( a n ) x n
F
a F
p ( x ) F [ x ]
p ( a )
p ( x )
x a
p ( x ) = a n x n + a n 1 x n 1 + + a 0 Z [ x ]
a n   0
p ( r / s ) = 0
gcd ( r , s ) = 1
r a 0
s a n
Q
Q
( Z [ x ] , + )
Φ n ( x ) = x n 1 x 1 = x n 1 + x n 2 + + x + 1
Φ p ( x )
Q
p
Φ n ( x ) = x n 1 x 1 = x n 1 + x n 2 + + x + 1
Φ p ( x )
Q
p
F
F [ x ]
R
R [ x ]
R
R [ x ]
x p x
p
Z p
p
x p x = x ( x 1 ) ( x 2 ) ( x ( p 1 ) )
F
f ( x ) = a 0 + a 1 x + + a n x n
F [ x ]
f ( x ) = a 1 + 2 a 2 x + + n a n x n 1
f ( x )
( f + g ) ( x ) = f ( x ) + g ( x )
D : F [ x ] F [ x ]
D ( f ( x ) ) = f ( x )
D
char F = 0
D
char F = p
( f g ) ( x ) = f ( x ) g ( x ) + f ( x ) g ( x )
f ( x ) F [ x ]
f ( x ) = a ( x a 1 ) ( x a 2 ) ( x a n )
f ( x )
f ( x )
f ( x )
F
F [ x ]
F [ x ]
R
R [ x 1 , , x n ]
R
R [ x ]
R
R
p ( x )
q ( x )
R [ x ]
R
deg ( p ( x ) + q ( x ) ) max ( deg p ( x ) , deg q ( x ) )
a x 2 + b x + c = 0
x = b ± b 2 4 a c 2 a
Δ = b 2 4 a c
Δ > 0
Δ = 0
Δ < 0
x 3 + b x 2 + c x + d = 0
y 3 + p y + q = 0
x = y b / 3
ω = 1 + i 3 2 ω 2 = 1 i 3 2 ω 3 = 1
y = z p 3 z
y
y 3 + p y + q = 0
A
B
z 3
p 3 / 27
A B 3 = p / 3
z
A 3 , ω A 3 , ω 2 A 3 , B 3 , ω B 3 , ω 2 B 3
y
ω i q 2 + p 3 27 + q 2 4 3 + ω 2 i q 2 p 3 27 + q 2 4 3
i = 0 , 1 , 2
Δ = p 3 27 + q 2 4
y 3 + p y + q = 0
Δ = 0
Δ > 0
Δ < 0
x 3 4 x 2 + 11 x + 30 = 0
x 3 3 x + 5 = 0
x 3 3 x + 2 = 0
x 3 + x + 3 = 0
x 4 + a x 3 + b x 2 + c x + d = 0
y 4 + p y 2 + q y + r = 0
x = y a / 4
( y 2 + 1 2 z ) 2 = ( z p ) y 2 q y + ( 1 4 z 2 r )
( m y + k ) 2
q 2 4 ( z p ) ( 1 4 z 2 r ) = 0
z 3 p z 2 4 r z + ( 4 p r q 2 ) = 0
( y 2 + 1 2 z ) 2 = ( m y + k ) 2
x 4 x 2 3 x + 2 = 0
x 4 + x 3 7 x 2 x + 6 = 0
x 4 2 x 2 + 4 x 3 = 0
x 4 4 x 3 + 3 x 2 5 x + 2 = 0
a 2 + 4 a + 2 = 0
a 2 = 4 a 3 = a + 2
a 2
a + 2
Z p
p
n
Z p
F
F [ x ]
7 5 = 16 807
5
Z 7
x + x 5 + x + 4
x 3 3 x + 4
Z 5
Z p
n
Z p
Z 5
p = x 4 + 4 x 2 + 4 x + 2
x
Z 5
Z 5
729
p = x 3 + 2 x 2 + 2 x + 4
q = x 4 + 2 x 2
r ( x )
s ( x )
r ( x ) p ( x ) + s ( x ) q ( x )
Z [ x ]
Q
Z
Q
D
F
D
p / q Q
p
q
1 / 2 = 2 / 4 = 3 / 6
a b = c d
a d = b c
Q
Z × Z
p / q
( p , q )
( 3 , 7 )
3 / 7
Z × Z
5 / 0
( 5 , 0 )
( 3 , 6 )
( 2 , 4 )
1 / 2
( a , b )
( c , d )
a d = b c
D
S = { ( a , b ) : a , b D and b   0 }
S
( a , b ) ( c , d )
a d = b c
S
D
a b = b a
D
( a , b ) ( c , d )
a d = b c
c b = d a
( c , d ) ( a , b )
( a , b ) ( c , d )
( c , d ) ( e , f )
a d = b c
c f = d e
a d = b c
f
a f d = a d f = b c f = b d e = b e d
D
a f = b e
( a , b ) ( e , f )
S
F D
F D
Q
a b + c d = a d + b c b d ; a b c d = a c b d
F D
( a , b ) S
[ a , b ]
F D
[ a , b ] + [ c , d ] = [ a d + b c , b d ]
[ a , b ] [ c , d ] = [ a c , b d ]
F D
[ a 1 , b 1 ] = [ a 2 , b 2 ]
[ c 1 , d 1 ] = [ c 2 , d 2 ]
[ a 1 d 1 + b 1 c 1 , b 1 d 1 ] = [ a 2 d 2 + b 2 c 2 , b 2 d 2 ]
( a 1 d 1 + b 1 c 1 ) ( b 2 d 2 ) = ( b 1 d 1 ) ( a 2 d 2 + b 2 c 2 )
[ a 1 , b 1 ] = [ a 2 , b 2 ]
[ c 1 , d 1 ] = [ c 2 , d 2 ]
a 1 b 2 = b 1 a 2
c 1 d 2 = d 1 c 2
( a 1 d 1 + b 1 c 1 ) ( b 2 d 2 ) = a 1 d 1 b 2 d 2 + b 1 c 1 b 2 d 2 = a 1 b 2 d 1 d 2 + b 1 b 2 c 1 d 2 = b 1 a 2 d 1 d 2 + b 1 b 2 d 1 c 2 = ( b 1 d 1 ) ( a 2 d 2 + b 2 c 2 )
S
F D
[ a , b ] + [ c , d ] = [ a d + b c , b d ] [ a , b ] [ c , d ] = [ a c , b d ]
[ 0 , 1 ]
[ 1 , 1 ]
[ 0 , 1 ]
[ a , b ] + [ 0 , 1 ] = [ a 1 + b 0 , b 1 ] = [ a , b ]
[ 1 , 1 ]
[ a , b ] F D
a   0
[ b , a ]
F D
[ a , b ] [ b , a ] = [ 1 , 1 ]
[ b , a ]
[ a , b ]
[ a , b ]
[ a , b ]
F D
F D
F D
[ a , b ] [ e , f ] + [ c , d ] [ e , f ] = [ a e , b f ] + [ c e , d f ] = [ a e d f + b f c e , b d f 2 ] = [ a e d + b c e , b d f ] = [ a d e + b c e , b d f ] = ( [ a , b ] + [ c , d ] ) [ e , f ]
F D
D
D
D
F D
F D
D
F D
E
D
ψ : F D E
E
ψ ( a ) = a
a D
a
F D
D
F D
ϕ : D F D
ϕ ( a ) = [ a , 1 ]
a
b
D
ϕ ( a + b ) = [ a + b , 1 ] = [ a , 1 ] + [ b , 1 ] = ϕ ( a ) + ϕ ( b )
ϕ ( a b ) = [ a b , 1 ] = [ a , 1 ] [ b , 1 ] = ϕ ( a ) ϕ ( b ) ;
ϕ
ϕ
ϕ ( a ) = ϕ ( b )
[ a , 1 ] = [ b , 1 ]
a = a 1 = 1 b = b
F D
D
ϕ ( a ) [ ϕ ( b ) ] 1 = [ a , 1 ] [ b , 1 ] 1 = [ a , 1 ] [ 1 , b ] = [ a , b ]
E
D
ψ : F D E
ψ ( [ a , b ] ) = a b 1
ψ
[ a 1 , b 1 ] = [ a 2 , b 2 ]
a 1 b 2 = b 1 a 2
a 1 b 1 1 = a 2 b 2 1
ψ ( [ a 1 , b 1 ] ) = ψ ( [ a 2 , b 2 ] )
[ a , b ]
[ c , d ]
F D
ψ ( [ a , b ] + [ c , d ] ) = ψ ( [ a d + b c , b d ] ) = ( a d + b c ) ( b d ) 1 = a b 1 + c d 1 = ψ ( [ a , b ] ) + ψ ( [ c , d ] )
ψ ( [ a , b ] [ c , d ] ) = ψ ( [ a c , b d ] ) = ( a c ) ( b d ) 1 = a b 1 c d 1 = ψ ( [ a , b ] ) ψ ( [ c , d ] )
ψ
ψ
ψ ( [ a , b ] ) = a b 1 = 0
a = 0 b = 0
[ a , b ] = [ 0 , b ]
ψ
[ 0 , b ]
F D
ψ
Q
Q [ x ]
Q [ x ]
p ( x ) / q ( x )
p ( x )
q ( x )
q ( x )
Q ( x )
Q
F
F
Q
F
p
F
Z p
F
F [ x ]
R
a
b
R
a
b
a b
c R
b = a c
R
a
b
R
u
R
a = u b
D
p D
p = a b
a
b
p
p a b
p a
p b
R
Q [ x , y ]
x 2
y 2
x y
R
x y
x y
x 2 y 2
x 2
y 2
n > 1
p 1 p k
p i
p i
D
D
a D
a   0
a
a
D
a = p 1 p r = q 1 q s
p i
q i
r = s
π S r
p i
q π ( j )
j = 1 , , r
Z [ 3 i ] = { a + b 3 i }
z = a + b 3 i
ν : Z [ 3 i ] N { 0 }
ν ( z ) = | z | 2 = a 2 + 3 b 2
ν ( z ) 0
z = 0
ν ( z w ) = ν ( z ) ν ( w )
ν ( z ) = 1
z
Z [ 3 i ]
1
1
4
4 = 2 2 = ( 1 3 i ) ( 1 + 3 i )
Z [ 3 i ]
2
2 = z w
z , w
Z [ 3 i ]
ν ( z ) = ν ( w ) = 2
z
Z [ 3 i ]
ν ( z ) = 2
a 2 + 3 b 2 = 2
2
1 3 i
1 + 3 i
2
1 3 i
1 + 3 i
4
R
a R
a = { r a : r R }
D
a , b D
a b
b a
a
b
b = a
a
D
a = D
a b
b = a x
x D
r
D
b r = ( a x ) r = a ( x r )
b a
b a
b a
b = a x
x D
a b
a
b
u
a = u b
b a
a b
b a
a = b
a = b
a b
b a
a = b x
b = a y
x , y D
a = b x = a y x
D
x y = 1
x
y
a
b
a D
a
1
a
1
a = 1 = D
D
p
D
p
p
p
a
D
p
p a
p
D = a
p = a
a
p
a
p
p
a
D
p a D
a p
p
a
a
p
D = a
p = a
p
D
p
p
p
p a b
a b p
p
p
a p
b p
p a
p b
D
I 1 , I 2 ,
I 1 I 2
N
I n = I N
n N
I = i = 1 I i
D
I
I 1 I
0 I
a , b I
a I i
b I j
i
j
N
i j
a
b
I j
a b
I j
r D
a I
a I i
i
I i
r a I i
I
I
D
D
a ¯ D
I
a ¯
I N
N N
I N = I = a ¯
I n = I N
n N
D
a
D
a
a = a 1 b 1
a 1
b 1
a a 1
a   a 1
a
a 1
b 1
a 1 = a 2 b 2
a 2
b 2
a 1 a 2
a a 1 a 2
N
a n = a N
n N
a N
a
a = c 1 p 1
p 1
c 1
a c 1
c 1
c 1 = c 2 p 2
p 2
c 2
a c 1 c 2
a = p 1 p 2 p r
p 1 , , p r
a = p 1 p 2 p r = q 1 q 2 q s
p i
q i
r < s
p 1
q 1 q 2 q s
q i
q i
p 1 q 1
q 1 = u 1 p 1
u 1
D
a = p 1 p 2 p r = u 1 p 1 q 2 q s
p 2 p r = u 1 q 2 q s
q i
p 2 = q 2 , p 3 = q 3 , , p r = q r
u 1 u 2 u r q r + 1 q s = 1
q r + 1 q s
q r + 1 , , q s
r = s
a
F
F [ x ]
Z [ x ]
Z [ x ]
I = { 5 f ( x ) + x g ( x ) : f ( x ) , g ( x ) Z [ x ] }
I
Z [ x ]
I = p ( x )
5 I
5 = f ( x ) p ( x )
p ( x ) = p
x I
x = p g ( x )
p = ± 1
p ( x ) = Z [ x ]
3
I
3 = 5 f ( x ) + x g ( x )
f ( x )
g ( x )
Z [ x ]
3 = 5 f ( x )
Z
F [ x ]
F
D
ν : D { 0 } N
a
a
b
D
ν ( a ) ν ( a b )
a , b D
b   0
q , r D
a = b q + r
r = 0
ν ( r ) < ν ( b )
D
ν
Z
F
F [ x ]
Z [ i ] = { a + b i : a , b Z }
a + b i
| a + b i | = a 2 + b 2
a 2 + b 2
ν ( a + b i ) = a 2 + b 2
ν ( a + b i ) = a 2 + b 2
Z [ i ]
z , w Z [ i ]
ν ( z w ) = | z w | 2 = | z | 2 | w | 2 = ν ( z ) ν ( w )
ν ( z ) 1
z Z [ i ]
ν ( z ) ν ( z ) ν ( w )
z = a + b i
w = c + d i
Z [ i ]
w   0
q
r
Z [ i ]
z = q w + r
r = 0
ν ( r ) < ν ( w )
z
w
Q ( i ) = { p + q i : p , q Q }
Z [ i ]
z w 1 = ( a + b i ) c d i c 2 + d 2 = a c + b d c 2 + d 2 + b c a d c 2 + d 2 i = ( m 1 + n 1 c 2 + d 2 ) + ( m 2 + n 2 c 2 + d 2 ) i = ( m 1 + m 2 i ) + ( n 1 c 2 + d 2 + n 2 c 2 + d 2 i ) = ( m 1 + m 2 i ) + ( s + t i )
Q ( i )
m i
| n i / ( a 2 + b 2 ) | 1 / 2
9 8 = 1 + 1 8 15 8 = 2 1 8
s
t
z w 1 = ( m 1 + m 2 i ) + ( s + t i )
s 2 + t 2 1 / 4 + 1 / 4 = 1 / 2
w
z = z w 1 w = w ( m 1 + m 2 i ) + w ( s + t i ) = q w + r
q = m 1 + m 2 i
r = w ( s + t i )
z
q w
Z [ i ]
r
Z [ i ]
r = 0
ν ( r ) < ν ( w )
ν ( r ) = ν ( w ) ν ( s + t i ) 1 2 ν ( w ) < ν ( w )
D
ν
D
I
D
b I
ν ( b )
a I
D
q
r
D
a = b q + r
r = 0
ν ( r ) < ν ( b )
r = a b q
I
I
r = 0
b
a = b q
I = b
D [ x ]
Z [ x ]
Z [ x ]
D
p ( x ) = a n x n + + a 1 x + a 0
D [ x ]
p ( x )
a 0 , , a n
p ( x )
gcd ( a 0 , , a n ) = 1
Z [ x ]
p ( x ) = 5 x 4 3 x 3 + x 4
1
q ( x ) = 4 x 2 6 x + 8
q ( x )
2
D
f ( x )
g ( x )
D [ x ]
f ( x ) g ( x )
f ( x ) = i = 0 m a i x i
g ( x ) = i = 0 n b i x i
p
f ( x ) g ( x )
r
p a r
s
p b s
x r + s
f ( x ) g ( x )
c r + s = a 0 b r + s + a 1 b r + s 1 + + a r + s 1 b 1 + a r + s b 0
p
a 0 , , a r 1
b 0 , , b s 1
p
c r + s
a r b s
p c r + s
p
a r
p
b s
D
p ( x )
q ( x )
D [ x ]
p ( x ) q ( x )
p ( x )
q ( x )
p ( x ) = c p 1 ( x )
q ( x ) = d q 1 ( x )
c
d
p ( x )
q ( x )
p 1 ( x )
q 1 ( x )
p ( x ) q ( x ) = c d p 1 ( x ) q 1 ( x )
p 1 ( x ) q 1 ( x )
p ( x ) q ( x )
c d
D
F
p ( x ) D [ x ]
p ( x ) = f ( x ) g ( x )
f ( x )
g ( x )
F [ x ]
p ( x ) = f 1 ( x ) g 1 ( x )
f 1 ( x )
g 1 ( x )
D [ x ]
deg f ( x ) = deg f 1 ( x )
deg g ( x ) = deg g 1 ( x )
a
b
D
a f ( x ) , b g ( x )
D [ x ]
a 1 , b 1 D
a f ( x ) = a 1 f 1 ( x )
b g ( x ) = b 1 g 1 ( x )
f 1 ( x )
g 1 ( x )
D [ x ]
a b p ( x ) = ( a 1 f 1 ( x ) ) ( b 1 g 1 ( x ) )
f 1 ( x )
g 1 ( x )
a b a 1 b 1
c D
p ( x ) = c f 1 ( x ) g 1 ( x )
deg f ( x ) = deg f 1 ( x )
deg g ( x ) = deg g 1 ( x )
D
F
p ( x )
D [ x ]
F [ x ]
D [ x ]
D
F
p ( x )
D [ x ]
p ( x ) = f ( x ) g ( x )
F [ x ]
p ( x ) = f 1 ( x ) g 1 ( x )
f 1 ( x )
g 1 ( x )
D [ x ]
deg f ( x ) = deg f 1 ( x )
deg g ( x ) = deg g 1 ( x )
D
D [ x ]
p ( x )
D [ x ]
p ( x )
D
p ( x )
D [ x ]
F
D
p ( x ) = f 1 ( x ) f 2 ( x ) f n ( x )
p ( x )
f i ( x )
a i D
a i f i ( x )
D [ x ]
b 1 , , b n D
a i f i ( x ) = b i g i ( x )
g i ( x )
D [ x ]
g i ( x )
D [ x ]
a 1 a n p ( x ) = b 1 b n g 1 ( x ) g n ( x )
b = b 1 b n
g 1 ( x ) g n ( x )
a 1 a n
b
p ( x ) = a g 1 ( x ) g n ( x )
a D
D
a
u c 1 c k
u
c i
D
p ( x ) = a 1 a m f 1 ( x ) f n ( x ) = b 1 b r g 1 ( x ) g s ( x )
p ( x )
D [ x ]
f i
g i
F [ x ]
a i
b i
F
F [ x ]
n = s
g i ( x )
f i ( x )
g i ( x )
i = 1 , , n
c 1 , , c n
d 1 , , d n
D
( c i / d i ) f i ( x ) = g i ( x )
c i f i ( x ) = d i g i ( x )
f i ( x )
g i ( x )
c i
d i
D
a 1 a m = u b 1 b r
D
u
D
D
m = s
b i
a i
b i
i
F
F [ x ]
Z [ x ]
D
D [ x 1 , , x n ]
Z [ x ]
17
n
N = 2 2 n + 1
N
i
1
z = a + b 3 i
Z [ 3 i ]
a 2 + 3 b 2 = 1
z
Z [ 3 i ]
1
1
z 1 = 1 / ( a + b 3 i ) = ( a b 3 i ) / ( a 2 + 3 b 2 )
Z [ 3 i ]
a 2 + 3 b 2 = 1
a = ± 1 , b = 0
Z [ i ]
Z [ i ]
5
1 + 3 i
6 + 8 i
2
5 = i ( 1 + 2 i ) ( 2 + i )
6 + 8 i = i ( 1 + i ) 2 ( 2 + i ) 2
D
F D
F D
F D
F
1
D
D
D
F
F
Q
F
F [ x ]
F ( x )
p ( x ) / q ( x )
q ( x )
x
p ( x 1 , , x n )
q ( x 1 , , x n )
F [ x 1 , , x n ]
p ( x 1 , , x n ) / q ( x 1 , , x n )
F [ x 1 , , x n ]
F [ x 1 , , x n ]
F ( x 1 , , x n )
x 1 , , x n
p
Z p [ x ]
Z p ( x )
Z p ( x )
p
Z [ i ]
Q ( i ) = { p + q i : p , q Q }
z = a + b i
w = c + d i   0
Z [ i ]
z / w Q ( i )
F
E
F
E
E
F
F
F
Q
F
p
F
Z p
Z [ 2 ] = { a + b 2 : a , b Z }
Z [ 2 ]
Z [ 2 ]
Z [ 2 ]
Z [ 2 i ]
ν ( a + b 2 i ) = a 2 + 2 b 2
D
d D
a
b
D
d a
d b
d
a
b
D
a
b
D
a
b
d
d
a
b
d
d
gcd ( a , b )
a
b
D
a
b
D
s
t
D
gcd ( a , b ) = a s + b t
D
D
a b
a
b
D
D
D
ν
u
D
ν ( u ) = ν ( 1 )
D
ν
a
b
D
ν ( a ) = ν ( b )
a = u b
u
ν ( b ) ν ( u b ) ν ( a )
ν ( a ) ν ( b )
Z [ 5 i ]
R
a 1 , , a n
R
r
a 1 r 1 + + a n r n
r 1 , , r n
R
R
R
D
I 1 I 2 I 3
N
I k = I N
k N
D
R
R
S
1 S
a b S
a , b S
R × S
( a , s ) ( a , s )
s S
s ( s a s a ) = 0
R × S
a / s
( a , s ) R × S
S 1 R
S 1 R
a s + b t = a t + b s s t a s b t = a b s t
S 1 R
S 1 R
S 1 R
R
S
ψ : R S 1 R
ψ ( a ) = a / 1
R
0 S
ψ
P
R
S = R P
R
P
R
S = R P
S 1 R
p
Z p
4
Z [ 3 i ]
N
Z
Q
R
X
X × X
P
X
X
( a , a ) P
a X
( a , b ) P
( b , a ) P
a = b
( a , b ) P
( b , c ) P
( a , c ) P
a b
( a , b ) P
a b
a
b
A B
A
B
X
a
b
a b
a
b
Z
X
X
X
X
P ( X )
X = { a , b , c }
P ( X )
{ a , b , c }
{ a } { b } { c } { a , b } { a , c } { b , c } { a , b , c } .
X
{ a , b , c }
P ( { a , b , c } )
G
G
N
a b
a b
a a
a N
m n
n m
m = n
m n
n p
m p
X = { 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24 }
24
X
24
Y
X
u
X
Y
a u
a Y
u
Y
u v
v
Y
u
Y
l
X
Y
l a
a Y
l
Y
k l
k
Y
l
Y
Y = { 2 , 3 , 4 , 6 }
X
Y
12
24
12
1
Y
X
Y
Y
Y
Y
u 1
u 2
Y
u 1 u
u
Y
u 1 u 2
u 2 u 1
u 1 = u 2
L
L
a , b L
a
b
a b
a
b
a , b L
a
b
a b
a
b
X
X
P ( X )
A
B
P ( X )
A
B
A B
A B
A
B
A A B
B A B
C
A
B
C
A B
A B
A
B
A
B
A B
G
X
G
X
G
H
K
G
H
K
H K
H K
G
H
K
H K
( A B )
A B
L
a , b , c L
a b = b a
a b = b a
a ( b c ) = ( a b ) c
a ( b c ) = ( a b ) c
a a = a
a a = a
a ( a b ) = a
a ( a b ) = a
a b
{ a , b }
b a
{ b , a }
{ a , b } = { b , a }
a ( b c )
( a b ) c
{ a , b , c }
d = a b
c d c = ( a b ) c
a a b = d d c = ( a b ) c
b ( a b ) c
( a b ) c
{ a , b , c }
( a b ) c
{ a , b , c }
u
{ a , b , c }
a u
b u
d = a b u
c u
( a b ) c = d c u
( a b ) c
{ a , b , c }
a ( b c )
{ a , b , c }
a ( b c ) = ( a b ) c
a
a
{ a }
a a = a
d = a b
a a d
d = a b a
a d a
a ( a b ) = a
L
L
L
a b
a b = b
L
a , b L
a
b
a b
a b
L
a a = a
a a
a b
b a
a b = b
b a = a
b = a b = b a = a
a b
b c
a b = b
b c = c
a c = a ( b c ) = ( a b ) c = b c = c
a c
L
a b
a b
a
b
a = ( a b ) a = a ( a b )
a a b
b a b
a b
a
b
u
a
b
a u
b u
a b u
( a b ) u = a ( b u ) = a u = u
a b
a
b
P ( X )
X
P ( X )
X
A
P ( X )
A X = A
A = A
I
X
a I
a X
O
X
O a
a X
A
P ( X )
A
A = X A = { x : x X and x A }
A A = X
A A =
L
I
O
a L
a
a a = I
a a = O
a
L
a ( b c ) = ( a b ) ( a c ) ;
P ( X )
A ( B C ) = ( A B ) ( A C )
A , B , C P ( X )
L
a ( b c ) = ( a b ) ( a c )
a , b , c L
L
a ( b c ) = ( a b ) ( a c )
a , b , c L
L
a ( b c ) = [ a ( a c ) ] ( b c ) = a [ ( a c ) ( b c ) ] = a [ ( c a ) ( c b ) ] = a [ c ( a b ) ] = a [ ( a b ) c ] = [ ( a b ) a ] [ ( a b ) c ] = ( a b ) ( a c )
B
I
O
B
X
P ( X )
B
B
a b = b a
a b = b a
a , b B
a ( b c ) = ( a b ) c
a ( b c ) = ( a b ) c
a , b , c B
a ( b c ) = ( a b ) ( a c )
a ( b c ) = ( a b ) ( a c )
a , b , c B
I
O
a O = a
a I = a
a B
a B
a B
a a = I
a a = O
B
a = a O = a ( a a ) = ( a a ) ( a a ) = ( a a ) I = a a
I b = ( b b ) b = ( b b ) b = b ( b b ) = b b = I
a ( a b ) = ( a I ) ( a b ) = a ( I b ) = a I = a
B
B
B
a B
O a = a
O a
O
B
I
B
a b = b
a b = a
a I = a
a B
a I = ( a I ) I = I ( I a ) = I
a I
a
B
B
B
B
I
O
B
a b
a b
{ a , b }
B
B
a I = I
a O = O
a B
a b = a c
a b = a c
a , b , c B
b = c
a b = I
a b = O
b = a
( a ) = a
a B
I = O
O = I
( a b ) = a b
( a b ) = a b
a b = a c
a b = a c
b = b ( b a ) = b ( a b ) = b ( a c ) = ( b a ) ( b c ) = ( a b ) ( b c ) = ( a c ) ( b c ) = ( c a ) ( c b ) = c ( a b ) = c ( a c ) = c ( c a ) = c
B
C
ϕ : B C
ϕ ( a b ) = ϕ ( a ) ϕ ( b ) ϕ ( a b ) = ϕ ( a ) ϕ ( b )
a
b
B
X
B
a B
B
a   O
a b = a
b B
b   O
a
B
b B
b   O
a
O b a
B
b
B
b   O
a
B
a b
b
a = b
b 1
O
b
b 1 b
b
b 1
b 2
O
b 1
b 2 b 1
b 2
a = b 2
O b 3 b 2 b 1 b
B
k
b k
a = b k
a
b
B
a   b
a b = O
a b
a
b
a b a
a b = a
a b = O
a b = a
a b
a = O
a
b
a b = O
B
a , b B
a b
a b = O
a b = I
a b
a b = b
a b = a ( a b ) = a ( a b ) = ( a a ) b = O b = O
a b = O
a b = ( a b ) = O = I
a b = I
a = a ( a b ) = ( a a ) ( a b ) = O ( a b ) = a b
a b
B
b
c
B
b ̸ c
a B
a b
a ̸ c
b c   O
a
a b c
a b
a ̸ c
b B
a 1 , , a n
B
a i b
b = a 1 a n
a , a 1 , , a n
B
a b
a i b
b = a a 1 a n
a = a i
i = 1 , , n
b 1 = a 1 a n
a i b
i
b 1 b
b b 1
b ̸ b 1
a
a b
a ̸ b 1
a
a b
a = a i
a i
a b 1
b b 1
b = a 1 a n
a
b
a = a b = a ( a 1 a n ) = ( a a 1 ) ( a a n )
O
a
a a i
a i
a = a i
i
B
X
B
P ( X )
B
P ( X )
X
B
a B
a
a = a 1 a n
a 1 , , a n X
ϕ : B P ( X )
ϕ ( a ) = ϕ ( a 1 a n ) = { a 1 , , a n }
ϕ
a = a 1 a n
b = b 1 b m
B
a i
b i
ϕ ( a ) = ϕ ( b )
{ a 1 , , a n } = { b 1 , , b m }
a = b
ϕ
a
b
ϕ
ϕ ( a b ) = ϕ ( a 1 a n b 1 b m ) = { a 1 , , a n , b 1 , , b m } = { a 1 , , a n } { b 1 , , b m } = ϕ ( a 1 a n ) ϕ ( b 1 b m ) = ϕ ( a ) ϕ ( b )
ϕ ( a b ) = ϕ ( a ) ϕ ( b )
2 n
n
a
a
b
A
B
a
b
a b
a
b
A
B
a
b
a b
a b
a b
a b
b a
a ( b c ) = ( a b ) ( a c )
a
a
a
a
I
O
a a = O
a a = I
a ( b c ) = ( a b ) ( a c )
a a = O
a a = I
( a b ) ( a b ) ( a b )
( a b ) ( a b ) ( a b ) = ( a b ) ( a b ) ( a b ) = ( a b ) ( a b ) = a ( b b ) = a O = a
a
( a b ) ( a b ) ( a b )
X = { a , b , c , d }
30
Z 12
B
210
B
a b
a b
B
X
B
P ( X )
B
Z
a b
a b
( a b a ) a
( a b ) ( a b )
a ( a b )
( c a b ) c ( a b )
( a b a ) a
a ( a b )
a
b
c
X
n
| P ( X ) | = 2 n
2 n
n N
a [ ( a b ) b ] = a ( a b )
a b
a b
L
L
a b
a b = b
a
b
a b
G
X
G
H
K
G
H
K
H K
R
X
R
X
I
J
X
I J
I
J
I + J
R
I , J
R
I + J = { r + s : r I and s J }
R
I
J
r 1 , r 2 I
s 1 , s 2 J
( r 1 + s 1 ) + ( r 2 + s 2 ) = ( r 1 + r 2 ) + ( s 1 + s 2 )
I + J
a R
a ( r 1 + s 1 ) = a r 1 + a s 1 I + J
I + J
R
B
a I = I
a O = O
a B
a b = I
a b = O
b = a
( a ) = a
a B
I = O
O = I
( a b ) = a b
( a b ) = a b
B
+
B
a + b = ( a b ) ( a b ) a b = a b
B
a 2 = a
a B
X
a
b
X
a b
b a
X
a b
N
N
Z
Q
R
X
Y
ϕ : X Y
a b
ϕ ( a ) ϕ ( b )
L
M
ψ : L M
ψ ( a b ) = ψ ( a ) ψ ( b )
ψ ( a b ) = ψ ( a ) ψ ( b )
B
a = b
( a b ) ( a b ) = O
a , b B
( )
a = b ( a b ) ( a b ) = ( a a ) ( a a ) = O O = O
( )
( a b ) ( a b ) = O a b = ( a a ) b = a ( a b ) = a [ I ( a b ) ] = a [ ( a a ) ( a b ) ] = [ a ( a b ) ] [ a ( a b ) ] = a [ ( a b ) ( a b ) ] = a 0 = a
a b = b
B
a = O
( a b ) ( a b ) = b
b B
L
M
L × M
( a , b ) ( c , d )
a c
b d
L × M
n
f : { O , I } n { 0 , I }
x 1 , , x n
O
I
x
y
x
x y
x y
0
0
1
0
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
X
n
n
n
C 1
C 2
C 2
C 1
C 1 = [ 2 , 1 , 2 ] [ 3 , 2 ] = C 2
n
5
1
72 = 2 3 3 2
n
16
4
16
5
x
y
z
n
V
F
α v
α v
α F
v V
α ( β v ) = ( α β ) v
( α + β ) v = α v + β v
α ( u + v ) = α u + α v
1 v = v
α , β F
u , v V
V
F
0
n
R n
R
u = ( u 1 , , u n )
v = ( v 1 , , v n )
R n
α
R
u + v = ( u 1 , , u n ) + ( v 1 , , v n ) = ( u 1 + v 1 , , u n + v n )
α u = α ( u 1 , , u n ) = ( α u 1 , , α u n )
F
F [ x ]
F
F [ x ]
α F
p ( x ) F [ x ]
α p ( x )
[ a , b ]
R
f ( x )
g ( x )
[ a , b ]
( f + g ) ( x )
f ( x ) + g ( x )
( α f ) ( x ) = α f ( x )
α R
f ( x ) = sin x
g ( x ) = x 2
( 2 f + 5 g ) ( x ) = 2 sin x + 5 x 2
V = Q ( 2 ) = { a + b 2 : a , b Q }
V
Q
u = a + b 2
v = c + d 2
u + v = ( a + c ) + ( b + d ) 2
V
α Q
α v
V
V
V
F
0 v = 0
v V
α 0 = 0
α F
α v = 0
α = 0
v = 0
( 1 ) v = v
v V
( α v ) = ( α ) v = α ( v )
α F
v V
0 v = ( 0 + 0 ) v = 0 v + 0 v ;
0 + 0 v = 0 v + 0 v
V
0 = 0 v
α = 0
α   0
α v = 0
1 / α
v = 0
v + ( 1 ) v = 1 v + ( 1 ) v = ( 1 1 ) v = 0 v = 0
v = ( 1 ) v
V
F
W
V
W
V
u , v W
α F
u + v
α v
W
W
R 3
W = { ( x 1 , 2 x 1 + x 2 , x 1 x 2 ) : x 1 , x 2 R }
W
R 3
α ( x 1 , 2 x 1 + x 2 , x 1 x 2 ) = ( α x 1 , α ( 2 x 1 + x 2 ) , α ( x 1 x 2 ) ) = ( α x 1 , 2 ( α x 1 ) + α x 2 , α x 1 α x 2 )
W
W
u = ( x 1 , 2 x 1 + x 2 , x 1 x 2 )
v = ( y 1 , 2 y 1 + y 2 , y 1 y 2 )
W
u + v = ( x 1 + y 1 , 2 ( x 1 + y 1 ) + ( x 2 + y 2 ) , ( x 1 + y 1 ) ( x 2 + y 2 ) )
W
F [ x ]
p ( x )
q ( x )
p ( x ) + q ( x )
α p ( x ) W
α F
p ( x ) W
V
F
v 1 , v 2 , , v n
V
α 1 , α 2 , , α n
F
w
V
w = i = 1 n α i v i = α 1 v 1 + α 2 v 2 + + α n v n
v 1 , v 2 , , v n
v 1 , v 2 , , v n
v 1 , v 2 , , v n
W
v 1 , v 2 , , v n
W
v 1 , v 2 , , v n
S = { v 1 , v 2 , , v n }
V
S
V
u
v
S
v i
u = α 1 v 1 + α 2 v 2 + + α n v n v = β 1 v 1 + β 2 v 2 + + β n v n
u + v = ( α 1 + β 1 ) v 1 + ( α 2 + β 2 ) v 2 + + ( α n + β n ) v n
v i
α F
α u = ( α α 1 ) v 1 + ( α α 2 ) v 2 + + ( α α n ) v n
S
S = { v 1 , v 2 , , v n }
V
α 1 , α 2 α n F
α i
α 1 v 1 + α 2 v 2 + + α n v n = 0
S
S
S
α 1 v 1 + α 2 v 2 + + α n v n = 0
α 1 = α 2 = = α n = 0
{ α 1 , α 2 α n }
{ v 1 , v 2 , , v n }
v = α 1 v 1 + α 2 v 2 + + α n v n = β 1 v 1 + β 2 v 2 + + β n v n
α 1 = β 1 , α 2 = β 2 , , α n = β n
v = α 1 v 1 + α 2 v 2 + + α n v n = β 1 v 1 + β 2 v 2 + + β n v n
( α 1 β 1 ) v 1 + ( α 2 β 2 ) v 2 + + ( α n β n ) v n = 0
v 1 , , v n
α i β i = 0
i = 1 , , n
{ v 1 , v 2 , , v n }
V
v i
{ v 1 , v 2 , , v n }
α 1 , , α n
α 1 v 1 + α 2 v 2 + + α n v n = 0
α i
α k   0
v k = α 1 α k v 1 α k 1 α k v k 1 α k + 1 α k v k + 1 α n α k v n
v k = β 1 v 1 + + β k 1 v k 1 + β k + 1 v k + 1 + + β n v n
β 1 v 1 + + β k 1 v k 1 v k + β k + 1 v k + 1 + + β n v n = 0
V
n
m > n
m
V
{ e 1 , e 2 , , e n }
V
V
{ e 1 , e 2 , , e n }
V
e 1 = ( 1 , 0 , 0 )
e 2 = ( 0 , 1 , 0 )
e 3 = ( 0 , 0 , 1 )
R 3
R 3
( x 1 , x 2 , x 3 )
R 3
x 1 e 1 + x 2 e 2 + x 3 e 3
e 1 , e 2 , e 3
e 1 , e 2 , e 3
R 3
{ ( 3 , 2 , 1 ) , ( 3 , 2 , 0 ) , ( 1 , 1 , 1 ) }
R 3
Q ( 2 ) = { a + b 2 : a , b Q }
{ 1 , 2 }
{ 1 + 2 , 1 2 }
Q ( 2 )
R 3
Q ( 2 )
{ e 1 , e 2 , , e m }
{ f 1 , f 2 , , f n }
V
m = n
{ e 1 , e 2 , , e m }
n m
{ f 1 , f 2 , , f n }
m n
m = n
{ e 1 , e 2 , , e n }
V
V
n
dim V = n
V
V
n
S = { v 1 , , v n }
V
S
V
S = { v 1 , , v n }
V
S
V
S = { v 1 , , v k }
V
k < n
v k + 1 , , v n
{ v 1 , , v k , v k + 1 , , v n }
V
V = Q ( 11 ) = { a + b 11 a , b Q }
S = { u }
u = 3 + 2 7 11 V
F
F [ x ]
F
F [ x ]
α p ( x )
α F
Q ( 2 )
Q ( 2 , 3 )
a + b 2 + c 3 + d 6
a , b , c , d
Q
Q ( 2 , 3 )
4
Q
Q ( 2 , 3 )
Q ( 2 , 3 )
{ 1 , 2 , 3 , 6 }
Q
2
R
P n
n
F [ x ]
P n
P n
{ 1 , x , x 2 , , x n 1 }
P n
F
n
F
F n
u = ( u 1 , , u n )
v = ( v 1 , , v n )
F n
α
F
u + v = ( u 1 , , u n ) + ( v 1 , , v n ) = ( u 1 + v 1 , , u n + v n )
α u = α ( u 1 , , u n ) = ( α u 1 , , α u n )
F n
n
R 3
{ ( x 1 , x 2 , x 3 ) : 3 x 1 2 x 2 + x 3 = 0 }
{ ( x 1 , x 2 , x 3 ) : 3 x 1 + 4 x 3 = 0 , 2 x 1 x 2 + x 3 = 0 }
{ ( x 1 , x 2 , x 3 ) : x 1 2 x 2 + 2 x 3 = 2 }
{ ( x 1 , x 2 , x 3 ) : 3 x 1 2 x 2 2 = 0 }
2
{ ( 1 , 0 , 3 ) , ( 0 , 1 , 2 ) }
( x , y , z ) R 3
A x + B y + C z = 0 D x + E y + C z = 0
R 3
W
[ 0 , 1 ]
f ( 0 ) = 0
W
C [ 0 , 1 ]
V
F
( α v ) = ( α ) v = α ( v )
α F
v V
0 = α 0 = α ( v + v ) = α ( v ) + α v
α v = α ( v )
V
n
S = { v 1 , , v n }
V
S
V
S = { v 1 , , v n }
V
S
V
S = { v 1 , , v k }
V
k < n
v k + 1 , , v n
{ v 1 , , v k , v k + 1 , , v n }
V
0
v 0 = 0 , v 1 , , v n V
α 0   0 , α 1 , , α n F
α 0 v 0 + + α n v n = 0
V
{ 0 }
V
V
n
m
V
m > n
V
W
F
m
n
T : V W
T ( u + v ) = T ( u ) + T ( v ) T ( α v ) = α T ( v )
α F
u , v V
T
V
W
T
ker ( T ) = { v V : T ( v ) = 0 }
V
T
T
T
R ( V ) = { w W : T ( v ) = w for some v V }
W
T : V W
ker ( T ) = { 0 }
{ v 1 , , v k }
T
{ v 1 , , v k , v k + 1 , , v m }
V
{ T ( v k + 1 ) , , T ( v m ) }
T
T
m k
dim V = dim W
T : V W
u , v ker ( T )
α F
T ( u + v ) = T ( u ) + T ( v ) = 0 T ( α v ) = α T ( v ) = α 0 = 0
u + v , α v ker ( T )
ker ( T )
V
T ( u ) = T ( v )
T ( u v ) = T ( u ) T ( v ) = 0
u v = 0
u = v
V
W
n
F
T : V W
{ v 1 , , v n }
V
{ T ( v 1 ) , , T ( v n ) }
W
F
n
F n
U
V
W
U
V
U + V
u + v
u U
v V
U + V
U V
W
U + V = W
U V = 0
W
W = U V
U
V
w W
w = u + v
u U
v V
U
k
W
n
V
n k
W = U V
V
U
V
W
dim ( U + V ) = dim U + dim V dim ( U V )
u , u U
v , v V
( u + v ) + ( u + v ) = ( u + u ) + ( v + v ) U + V α ( u + v ) = α u + α v U + V
V
W
F
V
W
Hom ( V , W )
F
U
V
( S + T ) ( v ) = S ( v ) + T ( v ) ( α S ) ( v ) = α S ( v )
S , T Hom ( V , W )
α F
v V
V
F
V
V = Hom ( V , F )
V
V
v 1 , , v n
V
v = α 1 v 1 + + α n v n
V
ϕ i : V F
ϕ i ( v ) = α i
ϕ i
V
v 1 , , v n
{ ( 3 , 1 ) , ( 2 , 2 ) }
R 2
( R 2 )
V
n
F
V
V
v V
λ v
V
v λ v
V
V
P n
n
F
F n
7 6 = 117 649
Z 7
{ 1 , a , a 2 , a 3 , a 4 , a 5 }
( Z 7 ) 6
( Z 7 ) 6
U
W
V
U + W
U + W = { u + w u U , w W }
U
W
U
W
V
5
6
2
U
W
U W
U + W
8
Q [ 2 4 ]
4
c = 2 4
F
p n
m × m
m
p
n
2 × 2
3 × 3
2 , 3 , 4 , 5
p
n
p n
F m
F
5 3
Z 5
a
F
M
3 × 3
Z 5
x F
{ 1 , a , a 2 }
M
M
F
{ 1 , a , a 2 }
R
R
F
F
M
a
a a 5
M
F
Q
R
F
p ( x ) F [ x ]
E
F
p ( x )
E [ x ]
p ( x ) = x 4 5 x 2 + 6
Q [ x ]
p ( x )
( x 2 2 ) ( x 2 3 )
Q [ x ]
p ( x )
p ( x ) = ( x 2 ) ( x + 2 ) ( x 3 ) ( x + 3 )
p ( x )
Q ( 2 ) = { a + b 2 : a , b Q }
F
E
F
F
E
F
F E
F = Q ( 2 ) = { a + b 2 : a , b Q }
E = Q ( 2 + 3 )
Q
2 + 3
E
F
E
F
2
E
2 + 3
E
1 / ( 2 + 3 ) = 3 2
E
2 + 3
3 2
2
3
E
p ( x ) = x 2 + x + 1 Z 2 [ x ]
p ( x )
Z 2
Z 2
α
p ( α ) = 0
p ( x )
p ( x )
Z 2 [ x ] / p ( x )
f ( x ) + p ( x )
Z 2 [ x ] / p ( x )
f ( x ) = ( x 2 + x + 1 ) q ( x ) + r ( x )
r ( x )
x 2 + x + 1
f ( x ) + x 2 + x + 1 = r ( x ) + x 2 + x + 1
r ( x )
0
1
x
1 + x
E = Z 2 [ x ] / x 2 + x + 1
Z 2
α
p ( x )
Z 2 ( α )
0 + 0 α = 0 1 + 0 α = 1 0 + 1 α = α 1 + 1 α = 1 + α
α 2 + α + 1 = 0
( 1 + α ) 2
( 1 + α ) ( 1 + α ) = 1 + α + α + ( α ) 2 = α
E
Z 2 ( α )
+ 0 1 α 1 + α 0 0 1 α 1 + α 1 1 0 1 + α α α α 1 + α 0 1 1 + α 1 + α α 1 0
Z 2 ( α )
0 1 α 1 + α 0 0 0 0 0 1 0 1 α 1 + α α 0 α 1 + α 1 1 + α 0 1 + α 1 α
F
p ( x )
F [ x ]
E
F
α E
p ( α ) = 0
p ( x )
E
F
α
p ( α ) = 0
p ( x )
p ( x )
F [ x ]
F [ x ] / p ( x )
E = F [ x ] / p ( x )
E
F
ψ : F F [ x ] / p ( x )
ψ ( a ) = a + p ( x )
a F
ψ
ψ ( a ) + ψ ( b ) = ( a + p ( x ) ) + ( b + p ( x ) ) = ( a + b ) + p ( x ) = ψ ( a + b )
ψ ( a ) ψ ( b ) = ( a + p ( x ) ) ( b + p ( x ) ) = a b + p ( x ) = ψ ( a b )
ψ
a + p ( x ) = ψ ( a ) = ψ ( b ) = b + p ( x )
a b
p ( x )
p ( x )
p ( x )
a b = 0
a = b
ψ
ψ
F
{ a + p ( x ) : a F }
E
E
F
p ( x )
α E
α = x + p ( x )
α
E
p ( x ) = a 0 + a 1 x + + a n x n
p ( α ) = a 0 + a 1 ( x + p ( x ) ) + + a n ( x + p ( x ) ) n = a 0 + ( a 1 x + p ( x ) ) + + ( a n x n + p ( x ) ) = a 0 + a 1 x + + a n x n + p ( x ) = 0 + p ( x )
α E = F [ x ] / p ( x )
α
p ( x )
p ( x ) = x 5 + x 4 + 1 Z 2 [ x ]
p ( x )
x 2 + x + 1
x 3 + x + 1
E
Z 2
p ( x )
E
E
Z 2 [ x ] / x 2 + x + 1
Z 2 [ x ] / x 3 + x + 1
Z 2 [ x ] / x 3 + x + 1
2 3 = 8
α
E
F
F
f ( α ) = 0
f ( x ) F [ x ]
E
F
F
E
F
F
E
F
E
F
α 1 , , α n
E
F
α 1 , , α n
F ( α 1 , , α n )
F
α 1 , , α n
E = F ( α )
α E
E
F
2
i
Q
x 2 2
x 2 + 1
π
e
Q
R
Q
Q
[ 0 , 1 ]
π + e
Q
C
Q
2 + 3
Q
α = 2 + 3
α 2 = 2 + 3
α 2 2 = 3
( α 2 2 ) 2 = 3
α 4 4 α 2 + 1 = 0
α
x 4 4 x 2 + 1 Q [ x ]
E
F
E
F
E
F
α E
α
F
F ( α )
F ( x )
F [ x ]
ϕ α : F [ x ] E
α
α
F
ϕ α ( p ( x ) ) = p ( α )   0
p ( x ) F [ x ]
ker ϕ α = { 0 }
ϕ α
E
F [ x ]
F [ x ]
F ( x )
E
E
F
α E
α
F
p ( x ) F [ x ]
p ( α ) = 0
f ( x )
F [ x ]
f ( α ) = 0
p ( x )
f ( x )
ϕ α : F [ x ] E
ϕ α
p ( x ) F [ x ]
deg p ( x ) 1
F [ x ]
α
p ( x )
F [ x ]
α
f ( α ) = 0
f ( x )
f ( x ) p ( x )
p ( x )
f ( x )
p ( x )
α
α
β p ( x )
β F
p ( x ) = r ( x ) s ( x )
p ( x )
p ( α ) = 0
r ( α ) s ( α ) = 0
r ( α ) = 0
s ( α ) = 0
p
p ( x )
E
F
α E
F
p ( x )
α
F
p ( x )
α
F
f ( x ) = x 2 2
g ( x ) = x 4 4 x 2 + 1
2
2 + 3
E
F
α E
F
F ( α ) F [ x ] / p ( x )
p ( x )
α
F
ϕ α : F [ x ] E
p ( x )
p ( x )
α
ϕ α
E
F ( α )
F
α
E = F ( α )
F
α E
F
α
F
n
β E
β = b 0 + b 1 α + + b n 1 α n 1
b i F
ϕ α ( F [ x ] ) F ( α )
E = F ( α )
ϕ α ( f ( x ) ) = f ( α )
f ( α )
α
F
p ( x ) = x n + a n 1 x n 1 + + a 0
α
p ( α ) = 0
α n = a n 1 α n 1 a 0
α n + 1 = α α n = a n 1 α n a n 2 α n 1 a 0 α = a n 1 ( a n 1 α n 1 a 0 ) a n 2 α n 1 a 0 α
α m
m n
α
n
β F ( α )
β = b 0 + b 1 α + + b n 1 α n 1
β = b 0 + b 1 α + + b n 1 α n 1 = c 0 + c 1 α + + c n 1 α n 1
b i
c i
F
g ( x ) = ( b 0 c 0 ) + ( b 1 c 1 ) x + + ( b n 1 c n 1 ) x n 1
F [ x ]
g ( α ) = 0
g ( x )
p ( x )
α
g ( x )
b 0 c 0 = b 1 c 1 = = b n 1 c n 1 = 0
b i = c i
i = 0 , 1 , , n 1
x 2 + 1
R
x 2 + 1
R [ x ]
E = R [ x ] / x 2 + 1
R
x 2 + 1
α = x + x 2 + 1
E
E
R ( α ) = { a + b α : a , b R }
α 2 = 1
E
α 2 + 1 = ( x + x 2 + 1 ) 2 + ( 1 + x 2 + 1 ) = ( x 2 + 1 ) + x 2 + 1 = 0
R ( α )
C
a + b α
a + b i
E
F
E
F
E
F
E
E
F
E
F
F
E = F ( α )
F
{ 1 , α , α 2 , , α n 1 }
E
F
F
n
E
n
F
[ E : F ] = n
E
F
E
F
E
F
α E
[ E : F ] = n
1 , α , , α n
a i F
a n α n + a n 1 α n 1 + + a 1 α + a 0 = 0
p ( x ) = a n x n + + a 0 F [ x ]
p ( α ) = 0
F
R
Q
Q
E
F
K
E
K
F
[ K : F ] = [ K : E ] [ E : F ]
{ α 1 , , α n }
E
F
{ β 1 , , β m }
K
E
{ α i β j }
K
F
K
u K
u = j = 1 m b j β j
b j = i = 1 n a i j α i
b j E
a i j F
u = j = 1 m ( i = 1 n a i j α i ) β j = i , j a i j ( α i β j )
m n
α i β j
K
F
{ α i β j }
v 1 , v 2 , , v n
V
c 1 v 1 + c 2 v 2 + + c n v n = 0
c 1 = c 2 = = c n = 0
u = i , j c i j ( α i β j ) = 0
c i j F
c i j
u
j = 1 m ( i = 1 n c i j α i ) β j = 0
i c i j α i E
β j
E
i = 1 n c i j α i = 0
j
α j
F
c i j = 0
i
j
F i
i = 1 , , k
F i + 1
F i
F k
F 1
[ F k : F 1 ] = [ F k : F k 1 ] [ F 2 : F 1 ]
E
F
α E
F
p ( x )
β F ( α )
q ( x )
deg q ( x )
deg p ( x )
deg p ( x ) = [ F ( α ) : F ]
deg q ( x ) = [ F ( β ) : F ]
F F ( β ) F ( α )
[ F ( α ) : F ] = [ F ( α ) : F ( β ) ] [ F ( β ) : F ]
Q
3 + 5
3 + 5
x 4 16 x 2 + 4
[ Q ( 3 + 5 ) : Q ] = 4
{ 1 , 3 }
Q ( 3 )
Q
3 + 5
Q ( 3 )
5
Q ( 3 )
{ 1 , 5 }
Q ( 3 , 5 ) = ( Q ( 3 ) ) ( 5 )
Q ( 3 )
{ 1 , 3 , 5 , 3 5 = 15 }
Q ( 3 , 5 ) = Q ( 3 + 5 )
Q
F ( α 1 , , α n )
F
n > 1
Q ( 5 3 , 5 i )
5
5
5 3
5
5 i Q ( 5 3 )
[ Q ( 5 3 , 5 i ) : Q ( 5 3 ) ] = 2
{ 1 , 5 i }
Q ( 5 3 , 5 i )
Q ( 5 3 )
{ 1 , 5 3 , ( 5 3 ) 2 }
Q ( 5 3 )
Q
Q ( 5 3 , 5 i )
Q
{ 1 , 5 i , 5 3 , ( 5 3 ) 2 , ( 5 6 ) 5 i , ( 5 6 ) 7 i = 5 5 6 i or 5 6 i }
5 6 i
x 6 + 5
Q
p = 5
Q Q ( 5 6 i ) Q ( 5 3 , 5 i )
Q ( 5 6 i ) = Q ( 5 3 , 5 i )
6
E
F
E
F
α 1 , , α n E
E = F ( α 1 , , α n )
E = F ( α 1 , , α n ) F ( α 1 , , α n 1 ) F ( α 1 ) F
F ( α 1 , , α i )
F ( α 1 , , α i 1 )
E
F
E
F
α 1 , , α n
E
E = F ( α 1 , , α n )
α i
F
E = F ( α 1 , , α n )
α i
F
E = F ( α 1 , , α n ) F ( α 1 , , α n 1 ) F ( α 1 ) F
F ( α 1 , , α i )
F ( α 1 , , α i 1 )
E = F ( α 1 , , α n ) F ( α 1 , , α n 1 ) F ( α 1 ) F
F ( α 1 , , α i )
F ( α 1 , , α i 1 )
F ( α 1 , , α i ) = F ( α 1 , , α i 1 ) ( α i )
α i
F ( α 1 , , α i 1 )
[ F ( α 1 , , α i ) : F ( α 1 , , α i 1 ) ]
i
[ E : F ]
F
E
p ( x )
E
E
F
E
F
α , β E
F
F ( α , β )
F
F ( α , β )
F
α ± β
α β
α / β
β   0
F
E
F
Q
E
F
F
E
E
F
F
F [ x ]
F
F
F [ x ]
F [ x ]
F
p ( x ) F [ x ]
p ( x )
F
α
x α
p ( x )
p ( x ) = ( x α ) q 1 ( x )
deg q 1 ( x ) = deg p ( x ) 1
q 1 ( x )
p ( x ) = ( x α ) ( x β ) q 2 ( x )
deg q 2 ( x ) = deg p ( x ) 2
p ( x )
p ( x )
F [ x ]
a x b
p ( b / a ) = 0
F
F
E
E
F
F E
α E
α
x α
α F
F = E
F
F
p ( x )
F [ x ]
F
p ( x )
E
F
p ( x )
F
p ( x )
p ( x )
F
p ( x ) = a 0 + a 1 x + + a n x n
F [ x ]
E
F
p ( x )
α 1 , , α n
E
E = F ( α 1 , , α n )
p ( x ) = ( x α 1 ) ( x α 2 ) ( x α n )
p ( x ) F [ x ]
E
E [ x ]
p ( x ) = x 4 + 2 x 2 8
Q [ x ]
p ( x )
x 2 2
x 2 + 4
Q ( 2 , i )
p ( x )
p ( x ) = x 3 3
Q [ x ]
p ( x )
Q ( 3 3 )
p ( x )
3 3 ± ( 3 6 ) 5 i 2
Q ( 3 3 )
p ( x ) F [ x ]
E
p ( x )
p ( x )
deg p ( x ) = 1
p ( x )
E = F
k
1 k < n
deg p ( x ) = n
p ( x )
K
p ( x )
α 1
K
p ( x ) = ( x α 1 ) q ( x )
q ( x ) K [ x ]
deg q ( x ) = n 1
E K
q ( x )
α 2 , , α n
p ( x )
E = K ( α 2 , , α n ) = F ( α 1 , , α n )
p ( x )
K
L
p ( x ) F [ x ]
ϕ : K L
F
ϕ : E F
K
E
α K
E
p ( x )
L
F
β
F [ x ]
p ( x )
ϕ
ϕ
ϕ ¯ : E ( α ) F ( β )
ϕ ¯ ( α ) = β
ϕ ¯
ϕ
E
p ( x )
n
E ( α )
1 , α , , α n 1
ϕ ¯ ( a 0 + a 1 α + + a n 1 α n 1 ) = ϕ ( a 0 ) + ϕ ( a 1 ) β + + ϕ ( a n 1 ) β n 1
a 0 + a 1 α + + a n 1 α n 1
E ( α )
ϕ ¯
ϕ ¯
ϕ
E [ x ]
F [ x ]
ϕ
ϕ ( a 0 + a 1 x + + a n x n ) = ϕ ( a 0 ) + ϕ ( a 1 ) x + + ϕ ( a n ) x n
ϕ : E F
ϕ ( p ( x ) ) = q ( x )
ϕ
p ( x )
q ( x )
ψ : E [ x ] / p ( x ) F [ x ] / q ( x )
σ : E [ x ] / p ( x ) E ( α )
τ : F [ x ] / q ( x ) F ( β )
α
β
ϕ ¯ = τ ψ σ 1
E [ x ] / p ( x ) ψ F [ x ] / q ( x ) σ τ E ( α ) ϕ ¯ F ( β ) E ϕ F
ϕ : E F
p ( x )
E [ x ]
q ( x )
F [ x ]
K
p ( x )
L
q ( x )
ϕ
ψ : K L
p ( x )
p ( x )
E
q ( x )
F
deg p ( x ) = 1
K = E
L = F
n
K
p ( x )
p ( x )
K
α
E E ( α ) K
β
q ( x )
L
F F ( β ) L
ϕ ¯ : E ( α ) F ( β )
ϕ ¯ ( α ) = β
ϕ ¯
ϕ
E
K ψ L σ τ E ( α ) ϕ ¯ F ( β ) E ϕ F
p ( x ) = ( x α ) f ( x )
q ( x ) = ( x β ) g ( x )
f ( x )
g ( x )
p ( x )
q ( x )
K
f ( x )
E ( α )
L
g ( x )
F ( β )
ψ : K L
ψ
ϕ ¯
E ( α )
ψ : K L
ψ
ϕ
E
p ( x )
F [ x ]
K
p ( x )
30
90
20
60
α
| α |
F
α
β
α + β
α β
α β
α / β
β   0
α
β
α > β
α + β
α β
α β
β > 1
A B C
A D E
α / 1 = x / β
x
α β
β < 1
α / β
β   0
α
α
A B D
B C D
A B C
1 / x = x / α
x 2 = α
P = ( p , q )
p
q
F
R
F
a x + b y + c = 0
a
b
c
F
F
F
x 2 + y 2 + d x + e y + f = 0
d
e
f
F
( x 1 , y 1 )
( x 2 , y 2 )
F
x 1 = x 2
x x 1 = 0
a x + b y + c = 0
x 1   x 2
y y 1 = ( y 2 y 1 x 2 x 1 ) ( x x 1 )
( x 1 , y 1 )
r
( x x 1 ) 2 + ( y y 1 ) 2 r 2 = 0
F
R
R
F
F
F
F
R
R
F
F
R
a x + b y + c = 0
F
F
x 2 + y 2 + d 1 x + e 1 y + f 1 = 0 x 2 + y 2 + d 2 x + e 2 y + f 2 = 0
d i
e i
f i
F
i = 1 , 2
x 2 + y 2 + d 1 x + e 1 x + f 1 = 0
( d 1 d 2 ) x + b ( e 2 e 1 ) y + ( f 2 f 1 ) = 0
a x + b y + c = 0 x 2 + y 2 + d x + e y + f = 0
y
A x 2 + B x + C = 0
A
B
C
F
x
x = B ± B 2 4 A C 2 A
F ( α )
α = B 2 4 A C > 0
F
F
F ( α )
α
F
α
Q = F 0 F 1 F k
F i = F i 1 ( α i )
α i F i
α F k
k > 0
[ Q ( α ) : Q ] = 2 k
F i
α i
[ F k : Q ] = [ F k : F k 1 ] [ F k 1 : F k 2 ] [ F 1 : Q ] = 2 k
Q
1
1
2
2 3
2 3
x 3 2
Q
[ Q ( 2 3 ) : Q ] = 3
3
2
1
π
π
π
π
20
60
cos 3 θ = cos ( 2 θ + θ ) = cos 2 θ cos θ sin 2 θ sin θ = ( 2 cos 2 θ 1 ) cos θ 2 sin 2 θ cos θ = ( 2 cos 2 θ 1 ) cos θ 2 ( 1 cos 2 θ ) cos θ = 4 cos 3 θ 3 cos θ
θ
α = cos θ
θ = 20
cos 3 θ = cos 60 = 1 / 2
4 α 3 3 α = 1 2
α
8 x 3 6 x 1
Z [ x ]
Q [ x ]
[ Q ( α ) : Q ] = 3
α
x 2 + y 2 = z 2
x n + y n = z n
n 3
p ( x , y )
Z [ x , y ]
E
F
F
α E
E
F
α
F
p ( x ) F [ x ]
Q
Q
1 / 3 + 7
3 + 5 3
3 + 2 i
cos θ + i sin θ
θ = 2 π / n
n N
2 3 i
x 4 ( 2 / 3 ) x 2 62 / 9
x 4 2 x 2 + 25
Q ( 3 , 6 )
Q
Q ( 2 3 , 3 3 )
Q
Q ( 2 , i )
Q
Q ( 3 , 5 , 7 )
Q
Q ( 2 , 2 3 )
Q
Q ( 8 )
Q ( 2 )
Q ( i , 2 + i , 3 + i )
Q
Q ( 2 + 5 )
Q ( 5 )
Q ( 2 , 6 + 10 )
Q ( 3 + 5 )
{ 1 , 2 , 3 , 6 }
{ 1 , i , 2 , 2 i }
{ 1 , 2 1 / 6 , 2 1 / 3 , 2 1 / 2 , 2 2 / 3 , 2 5 / 6 }
x 4 10 x 2 + 21
Q
x 4 + 1
Q
x 3 + 2 x + 2
Z 3
x 3 3
Q
Q ( 3 , 7 )
Q ( 3 4 , i )
Q
Q ( 3 4 , i )
Q
[ Q ( 3 4 , i ) : Q ] = 8
F
Q ( 3 4 , i )
[ F : Q ] = 2
F
Q ( 3 4 , i )
[ F : Q ] = 4
Z 2 [ x ] / x 3 + x + 1
Z 2 [ x ] / x 3 + x + 1
α
1 + α
α 2
1 + α 2
α + α 2
1 + α + α 2
α 3 + α + 1 = 0
9
20
cos 1
Q
Q ( 3 , 3 4 , 3 8 , )
Q
π
Q ( π 3 )
p ( x )
n
F [ x ]
E
p ( x )
[ E : F ] n !
Q ( 2 ) Q ( 3 )
Q ( 3 4 )
Q ( 3 4 i )
K
E
E
F
K
F
E
F
K
E
α K
α
F
α
E
p ( x ) = β 0 + β 1 x + + β n x n
E [ x ]
α
F ( β 0 , , β n )
Z [ x ] / x 3 2
F
p
p ( x ) = x p a
F
F
E
F
p ( x )
F [ x ]
E
p ( x )
F [ x ]
F
α
β
β   0
α / β
R
Q
Q
E
F
σ
E
F
α E
σ
α
E
Q ( 3 , 7 ) = Q ( 3 + 7 )
Q ( a , b ) = Q ( a + b )
gcd ( a , b ) = 1
{ 1 , 3 , 7 , 21 }
Q ( 3 , 7 )
Q
Q ( 3 , 7 ) Q ( 3 + 7 )
[ Q ( 3 , 7 ) : Q ] = 4
[ Q ( 3 + 7 ) : Q ] = 2
3 + 7
Q ( 3 , 7 ) = Q ( 3 + 7 )
E
F
[ E : F ] = 2
E
F
f ( x ) F [ x ]
p ( x )
Z 6 [ x ]
R
p ( x )
R
E
F
α E
[ F ( α ) : F ( α 3 ) ]
α , β
Q
α β
α + β
E
F
α E
F
F ( α )
F
F
β F ( α )
F
β = p ( α ) / q ( α )
p
q
α
q ( α )   0
F
β
F
f ( x ) F [ x ]
f ( β ) = 0
f ( x ) = a 0 + a 1 x + + a n x n
0 = f ( β ) = f ( p ( α ) q ( α ) ) = a 0 + a 1 ( p ( α ) q ( α ) ) + + a n ( p ( α ) q ( α ) ) n
q ( α ) n
F [ x ]
α
α
p ( x ) F [ x ]
deg p = n
[ F ( α ) : F ] = n
Q Q [ 3 ] Q [ 3 , 2 ]
2 3
2
3
Q
p ( x ) = x 4 + x 2 1
a 2 + 1
8
8
p ( x ) = x 4 + x 2 1
( w r )
60
20
2
p ( x ) = x 5 + 2 x 4 + 1
Z 3
p ( x )
Z 3
p ( x )
3 5
p ( x )
p ( x )
3 5 = 243
p ( x )
Z 3
p ( x )
p ( x )
r ( x ) = x 4 + 2 x + 2
p = 2
r ( x )
r ( x )
s ( x ) = x 4 + x 2 + 1
s ( x )
K
q ( x ) = x 3 + 3 x 2 + 3 x 2
q ( x )
K
K
K
q ( x )
L
M
q ( x )
P
q ( x )
P
q ( x )
Z p
p
p n
p
n
F
p
p
α
F
p α = 0
F
0
p
F
n
n α = 0
α
F
F
p
p
n
F
F
p
p
p
F
p
F
p n
n N
ϕ : Z F
ϕ ( n ) = n 1
F
p
ϕ
p Z
ϕ
F
Z p
K
F
K
K
[ F : K ] = n
F
F
K
α 1 , , α n F
α
F
α = a 1 α 1 + + a n α n
a i
K
p
K
p n
α i
F
p n
p
D
p
a p n + b p n = ( a + b ) p n
n
n
n = 1
( a + b ) p = k = 0 p ( p k ) a k b p k
0 < k < p
( p k ) = p ! k ! ( p k ) !
p
p
k ! ( p k ) !
D
p
( a + b ) p = a p + b p
k
1 k n
( a + b ) p n + 1 = ( ( a + b ) p ) p n = ( a p + b p ) p n = ( a p ) p n + ( b p ) p n = a p n + 1 + b p n + 1
n + 1
F
f ( x ) F [ x ]
n
n
f ( x )
f ( x )
f
E
F
F
E
F [ x ]
x 2 2
Q
( x 2 ) ( x + 2 )
Q ( 2 )
Q
α = a + b 2
Q ( 2 )
b = 0
α
x a
b   0
α
x 2 2 a x + a 2 2 b 2 = ( x ( a + b 2 ) ) ( x ( a b 2 ) )
f ( x ) = a 0 + a 1 x + + a n x n
F [ x ]
f ( x )
f ( x ) = a 1 + 2 a 2 x + + n a n x n 1
F
f ( x ) F [ x ]
f ( x )
f ( x )
f ( x )
f ( x )
f ( x )
F
f ( x ) = ( x α 1 ) ( x α 2 ) ( x α n )
α i   α j
i   j
f ( x )
f ( x ) = ( x α 2 ) ( x α n ) + ( x α 1 ) ( x α 3 ) ( x α n ) + + ( x α 1 ) ( x α n 1 )
f ( x )
f ( x )
f ( x ) = ( x α ) k g ( x )
k > 1
f ( x ) = k ( x α ) k 1 g ( x ) + ( x α ) k g ( x )
f ( x )
f ( x )
p
n
F
p n
p n
x p n x
Z p
f ( x ) = x p n x
F
f ( x )
f ( x )
p n
F
f ( x ) = p n x p n 1 1 = 1
f ( x )
f ( x )
F
f ( x )
α
β
f ( x )
α + β
α β
f ( x )
α p n + β p n = ( α + β ) p n
α p n β p n = ( α β ) p n
f ( x )
f ( x )
α
f ( x )
α
f ( x )
f ( α ) = ( α ) p n ( α ) = α p n + α = ( α p n α ) = 0
p
p = 2
f ( α ) = ( α ) 2 n ( α ) = α + α = 0
α   0
( α 1 ) p n = ( α p n ) 1 = α 1
f ( x )
F
f ( x )
F
E
p n
E
F
E
f ( x )
f ( x )
α
E
E
p n 1
α p n 1 = 1
α p n α = 0
E
p n
E
f ( x )
p n
p n
GF ( p n )
p n
GF ( p n )
p m
m
n
m n
m > 0
GF ( p n )
GF ( p m )
F
E = GF ( p n )
F
K
p m
K
Z p
m n
[ E : K ] = [ E : F ] [ F : K ]
m n
m > 0
p m 1
p n 1
x p m 1 1
x p n 1 1
x p m x
x p n x
x p m x
x p n x
GF ( p n )
x p m x
GF ( p m )
GF ( p 24 )
GF ( p 24 )
F
F
F
F
G
F
F
G
G
F
n
G Z p 1 e 1 × × Z p k e k
n = p 1 e 1 p k e k
p 1 , , p k
m
p 1 e 1 , , p k e k
G
m
α
G
x r 1
r
m
α
x m 1
x m 1
m
F
n m
m | G |
m = n
G
n
E
F
F
α
E
E
E = F ( α )
GF ( 2 4 )
Z 2 / 1 + x + x 4
GF ( 2 4 )
{ a 0 + a 1 α + a 2 α 2 + a 3 α 3 : a i Z 2 and 1 + α + α 4 = 0 }
1 + α + α 4 = 0
GF ( 2 4 )
GF ( 2 4 )
Z 15
α
α 1 = α α 6 = α 2 + α 3 α 11 = α + α 2 + α 3 α 2 = α 2 α 7 = 1 + α + α 3 α 12 = 1 + α + α 2 + α 3 α 3 = α 3 α 8 = 1 + α 2 α 13 = 1 + α 2 + α 3 α 4 = 1 + α α 9 = α + α 3 α 14 = 1 + α 3 α 5 = α + α 2 α 10 = 1 + α + α 2 α 15 = 1.
( n , k )
E : Z 2 k Z 2 n
D : Z 2 n Z 2 k
D
H M k × n ( Z 2 )
ϕ : Z 2 k Z 2 n
( n , k )
ϕ
( a 1 , a 2 , , a n )
n
( a n , a 1 , a 2 , , a n 1 )
( 6 , 3 )
G 1 = ( 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 ) and G 2 = ( 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 0 0 1 )
( 000 ) ( 000000 ) ( 100 ) ( 100100 ) ( 001 ) ( 001001 ) ( 101 ) ( 101101 ) ( 010 ) ( 010010 ) ( 110 ) ( 110110 ) ( 011 ) ( 011011 ) ( 111 ) ( 111111 ) .
( 000 ) ( 000000 ) ( 100 ) ( 111100 ) ( 001 ) ( 001111 ) ( 101 ) ( 110011 ) ( 010 ) ( 011110 ) ( 110 ) ( 100010 ) ( 011 ) ( 010001 ) ( 111 ) ( 101101 ) .
( 101101 )
( 011011 )
Z 2
n
Z 2 [ x ]
n
( a 0 , a 1 , , a n 1 )
f ( x ) = a 0 + a 1 x + + a n 1 x n 1
f ( x )
n 1
5
( 10011 )
1 + 0 x + 0 x 2 + 1 x 3 + 1 x 4 = 1 + x 3 + x 4
f ( x ) Z 2 [ x ]
deg f ( x ) < n
n
x + x 2 + x 4
5
( 01101 )
g ( x )
Z 2 [ x ]
n k
( n , k )
C
( a 0 , , a k 1 )
k
f ( x ) = a 0 + a 1 x + + a k 1 x k 1
Z 2 [ x ]
f ( x )
g ( x )
C
Z 2 [ x ]
n
g ( x )
g ( x ) = 1 + x 3
( 6 , 3 )
C
3
( a 0 , a 1 , a 2 )
f ( x ) = a 0 + a 1 x + a 2 x 2
1 + x 3
ϕ : Z 2 3 Z 2 6
ϕ : f ( x ) g ( x ) f ( x )
Z 2 n
Z 2
ϕ
ϕ
ϕ ( a 0 , a 1 , a 2 ) = ( 000000 )
0 + 0 x + 0 x 2 + 0 x 3 + 0 x 4 + 0 x 5 = ( 1 + x 3 ) ( a 0 + a 1 x + a 2 x 2 ) = a 0 + a 1 x + a 2 x 2 + a 0 x 3 + a 1 x 4 + a 2 x 5
a 0 + a 1 x + a 2 x 2
ker ϕ = { ( 000 ) }
ϕ
C
1
x
x 2
( 1 + x 3 ) 1 = 1 + x 3 ( 1 + x 3 ) x = x + x 4 ( 1 + x 3 ) x 2 = x 2 + x 5
G 1
H = ( 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 )
2
x n 1 = ( x 1 ) ( x n 1 + + x + 1 )
R n = Z 2 [ x ] / x n 1
f ( t ) = a 0 + a 1 t + + a n 1 t n 1
t n = 1
Z 2 n
R n
Z 2 n
Z [ x ] / x n 1
Z [ x ] / x n 1
n
f ( t ) = a 0 + a 1 t + + a n 1 t n 1
R n
t f ( t ) = a n 1 + a 0 t + + a n 2 t n 1
f ( t )
t
R n
C
Z 2 n
R n = Z [ x ] / x n 1
C
f ( t )
C
t f ( t )
C
t k f ( t )
C
k N
C
f ( t ) , t f ( t ) , t 2 f ( t ) , , t n 1 f ( t )
p ( t )
p ( t ) f ( t )
C
C
C
Z 2 [ x ] / x n + 1
f ( t ) = a 0 + a 1 t + + a n 1 t n 1
C
t f ( t )
C
( a 1 , , a n 1 , a 0 )
C
R n
Z 2 n
R n
ϕ : Z 2 [ x ] R n
ϕ [ f ( x ) ] = f ( t )
ϕ
x n 1
C
R n
ϕ ( I )
I
Z 2 [ x ]
x n 1
I
Z 2 [ x ]
Z 2
I = g ( x )
Z 2 [ x ]
x n 1
I
g ( x )
x n 1
C
R n
C = g ( t ) = { f ( t ) g ( t ) : f ( t ) R n and g ( x ) ( x n 1 ) in Z 2 [ x ] }
C
C
x 7 1
x 7 1 = ( 1 + x ) ( 1 + x + x 3 ) ( 1 + x 2 + x 3 )
g ( t ) = ( 1 + t + t 3 )
C
R 7
( 7 , 4 )
g ( t )
t
t 2
t 3
C
G = ( 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 1 )
( n , k )
C
t k
x n 1 = g ( x ) h ( x )
Z 2 [ x ]
g ( x ) = g 0 + g 1 x + + g n k x n k
h ( x ) = h 0 + h 1 x + + h k x k
n × k
G = ( g 0 0 0 g 1 g 0 0 g n k g n k 1 g 0 0 g n k g 1 0 0 g n k )
C
g ( t )
C
( n k ) × n
H = ( 0 0 0 h k h 0 0 0 h k h 0 0 h k h 0 0 0 0 )
C = g ( t )
R n
x n 1 = g ( x ) h ( x )
G
H
C
H G = 0
x 7 1 = g ( x ) h ( x ) = ( 1 + x + x 3 ) ( 1 + x + x 2 + x 4 )
H = ( 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 )
α 1 , , α n
F
n × n
( 1 1 1 α 1 α 2 α n α 1 2 α 2 2 α n 2 α 1 n 1 α 2 n 1 α n n 1 )
α 1 , , α n
F
n 2
det ( 1 1 1 α 1 α 2 α n α 1 2 α 2 2 α n 2 α 1 n 1 α 2 n 1 α n n 1 ) = 1 j < i n ( α i α j )
α i
n
n = 2
α 2 α 1
n 1
p ( x )
p ( x ) = det ( 1 1 1 1 α 1 α 2 α n 1 x α 1 2 α 2 2 α n 1 2 x 2 α 1 n 1 α 2 n 1 α n 1 n 1 x n 1 )
p ( x )
n 1
p ( x )
α 1 , , α n 1
p ( x ) = ( x α 1 ) ( x α 2 ) ( x α n 1 ) β
β = ( 1 ) n + n det ( 1 1 1 α 1 α 2 α n 1 α 1 2 α 2 2 α n 1 2 α 1 n 2 α 2 n 2 α n 1 n 2 )
β = ( 1 ) n + n 1 j < i n 1 ( α i α j )
x = α n
C = g ( t )
R n
ω
n
Z 2
s
ω
g ( x )
C
s + 1
g ( ω r ) = g ( ω r + 1 ) = = g ( ω r + s 1 ) = 0
f ( x )
C
s
f ( x ) = a i 0 x i 0 + a i 1 x i 1 + + a i s 1 x i s 1
C
a i
g ( ω r ) = g ( ω r + 1 ) = = g ( ω r + s 1 ) = 0
g ( x )
f ( x )
f ( ω r ) = f ( ω r + 1 ) = = f ( ω r + s 1 ) = 0
a i 0 ( ω r ) i 0 + a i 1 ( ω r ) i 1 + + a i s 1 ( ω r ) i s 1 = 0 a i 0 ( ω r + 1 ) i 0 + a i 1 ( ω r + 1 ) i 2 + + a i s 1 ( ω r + 1 ) i s 1 = 0 a i 0 ( ω r + s 1 ) i 0 + a i 1 ( ω r + s 1 ) i 1 + + a i s 1 ( ω r + s 1 ) i s 1 = 0
( a i 0 , a i 1 , , a i s 1 )
( ω i 0 ) r x 0 + ( ω i 1 ) r x 1 + + ( ω i s 1 ) r x n 1 = 0 ( ω i 0 ) r + 1 x 0 + ( ω i 1 ) r + 1 x 1 + + ( ω i s 1 ) r + 1 x n 1 = 0 ( ω i 0 ) r + s 1 x 0 + ( ω i 1 ) r + s 1 x 1 + + ( ω i s 1 ) r + s 1 x n 1 = 0
( ( ω i 0 ) r ( ω i 1 ) r ( ω i s 1 ) r ( ω i 0 ) r + 1 ( ω i 1 ) r + 1 ( ω i s 1 ) r + 1 ( ω i 0 ) r + s 1 ( ω i 1 ) r + s 1 ( ω i s 1 ) r + s 1 )
a i 0 = a i 1 = = a i s 1 = 0
231
24
231 + 24 = 255 = 2 8 1
( 255 , 231 )
1
16
d = 2 r + 1
r 0
ω
n
Z 2
m i ( x )
Z 2
ω i
g ( x ) = lcm [ m 1 ( x ) , m 2 ( x ) , , m 2 r ( x ) ]
g ( t )
R n
n
d
C
d
C = g ( t )
R n
C
d
f ( t )
C
f ( ω i ) = 0
1 i < d
H = ( 1 ω ω 2 ω n 1 1 ω 2 ω 4 ω ( n 1 ) ( 2 ) 1 ω 3 ω 6 ω ( n 1 ) ( 3 ) 1 ω 2 r ω 4 r ω ( n 1 ) ( 2 r ) )
C
f ( t )
C
g ( x ) f ( x )
Z 2 [ x ]
i = 1 , , 2 r
f ( ω i ) = 0
g ( ω i ) = 0
f ( ω i ) = 0
1 i d
f ( x )
m i ( x )
m i ( x )
ω i
g ( x ) f ( x )
g ( x )
f ( x )
f ( t ) = a 0 + a 1 t + + a n 1 v t n 1
R n
n
Z 2 n
x = ( a 0 a 1 a n 1 ) t
H x = ( a 0 + a 1 ω + + a n 1 ω n 1 a 0 + a 1 ω 2 + + a n 1 ( ω 2 ) n 1 a 0 + a 1 ω 2 r + + a n 1 ( ω 2 r ) n 1 ) = ( f ( ω ) f ( ω 2 ) f ( ω 2 r ) ) = 0
f ( t )
C
H
C
f ( t ) = a 0 + a 1 t + + a n 1 t n 1
C
f ( ω i ) = 0
i = 1 , , 2 r
g ( t ) = lcm [ m 1 ( t ) , , m 2 r ( t ) ]
C = g ( t )
x 15 1 Z 2 [ x ]
x 15 1 = ( x + 1 ) ( x 2 + x + 1 ) ( x 4 + x + 1 ) ( x 4 + x 3 + 1 ) ( x 4 + x 3 + x 2 + x + 1 )
ω
1 + x + x 4
GF ( 2 4 )
{ a 0 + a 1 ω + a 2 ω 2 + a 3 ω 3 : a i Z 2 and 1 + ω + ω 4 = 0 }
ω
15
ω
m 1 ( x ) = 1 + x + x 4
ω 2
ω 4
m 1 ( x )
ω 3
m 2 ( x ) = 1 + x + x 2 + x 3 + x 4
g ( x ) = m 1 ( x ) m 2 ( x ) = 1 + x 4 + x 6 + x 7 + x 8
ω
ω 2
ω 3
ω 4
m 1 ( x )
m 2 ( x )
x 15 1
( 15 , 7 )
x 15 1 = g ( x ) h ( x )
h ( x ) = 1 + x 4 + x 6 + x 7
( 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 )
[ GF ( 3 6 ) : GF ( 3 3 ) ]
[ GF ( 128 ) : GF ( 16 ) ]
[ GF ( 625 ) : GF ( 25 ) ]
[ GF ( p 12 ) : GF ( p 2 ) ]
[ GF ( p m ) : GF ( p n ) ]
n m
GF ( p 30 )
α
x 3 + x 2 + 1
Z 2
8
x 3 + x 2 + 1
Z 2 ( α )
Z 2 ( α )
x 3 + x 2 + 1
α
27
p ( x )
Z 3 [ x ]
3
Z 3 [ x ] / p ( x )
27
Q
Z 2 [ x ]
x 5 1
x 6 + x 5 + x 4 + x 3 + x 2 + x + 1
x 9 1
x 4 + x 3 + x 2 + x + 1
x 5 1 = ( x + 1 ) ( x 4 + x 3 + x 2 + x + 1 )
x 9 1 = ( x + 1 ) ( x 2 + x + 1 ) ( x 6 + x 3 + 1 )
Z 2 [ x ] / x 3 + x + 1 Z 2 [ x ] / x 3 + x 2 + 1
n
n = 6 , 7 , 8 , 10
t + 1
R n
Z 2 n
7
15
x 7 1 = ( x + 1 ) ( x 3 + x + 1 ) ( x 3 + x 2 + 1 )
p
Z p ( x )
p
D
p
( a b ) p n = a p n b p n
a , b D
E
F
K
E
F
E = F
F E K
K
F
K
E
p ( x ) F [ x ]
p ( x ) E [ x ]
E
F
F
q
α E
F
n
F ( α )
q n
α
F
n
β F ( α )
β = a 0 + a 1 α + + a n 1 α n 1
a i F
q n
n
( a 0 , a 1 , , a n 1 )
F
E
F
α E
E = F ( α )
n
n
Z p [ x ]
Φ : GF ( p n ) GF ( p n )
Φ : α α p
n
GF ( p n )
a p
a GF ( p n )
E
F
GF ( p n )
| E | = p r
| F | = p s
E F
p
( p 1 ) ! 1 ( mod p )
x p 1 1
Z p
g ( t )
C
R n
g ( x )
1
C
( n , k )
n k
R n
Z 2 n
C
R n
g ( t )
f ( t )
R n
g ( t ) f ( t )
f ( x )
g ( x )
Z 2 [ x ]
C = g ( t )
R n
x n 1 = g ( x ) h ( x )
g ( x ) = g 0 + g 1 x + + g n k x n k
h ( x ) = h 0 + h 1 x + + h k x k
G
n × k
G = ( g 0 0 0 g 1 g 0 0 g n k g n k 1 g 0 0 g n k g 1 0 0 g n k )
H
( n k ) × n
H = ( 0 0 0 h k h 0 0 0 h k h 0 0 h k h 0 0 0 0 )
G
C
H
C
H G = 0
C
R n
c ( t ) = c 0 + c 1 t + + c n 1 t n 1
w ( t ) = w 0 + w 1 t + w n 1 t n 1
R n
a 1 , , a k
w ( t ) = c ( t ) + e ( t )
e ( t ) = t a 1 + t a 2 + + t a k
a i
c ( t )
w ( t )
a i
w ( t )
w ( ω i ) = s i
i = 1 , , 2 r
ω
n
Z 2
w ( t )
s 1 , , s 2 r
w ( t )
s i = 0
i
s i = w ( ω i ) = e ( ω i ) = ω i a 1 + ω i a 2 + + ω i a k
i = 1 , , 2 r
s ( x ) = ( x + ω a 1 ) ( x + ω a 2 ) ( x + ω a k )
( 15 , 7 )
a 1
a 2
s ( x ) = ( x + ω a 1 ) ( x + ω a 2 )
s ( x ) = x 2 + s 1 x + ( s 1 2 + s 3 s 1 )
w ( t ) = 1 + t 2 + t 4 + t 5 + t 7 + t 12 + t 13
5 2
p ( x ) = x 25 x
2 7
0
3 6
3 2
2 | 6
8
8
9
Q ( 3 , 7 )
x 2 3
x 2 7
3
7
Q ( 3 , 7 )
p = 2 , 3 , 5 , 7
3 n 10
n
τ
E
K = { b E τ ( b ) = b }
E
τ
E
E = G F ( 3 6 )
p = 2
F
a F
{ x 2 | x F } { a x 2 | x F }
S e t ( )
a
p = 2
x 5 1 = 0
x 6 x 3 6 = 0
a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0
F
σ
τ
F
σ τ
σ 1
F
E
F
E
F
σ : E E
σ ( α ) = α
α F
E
F
E
σ
τ
E
σ ( α ) = α
τ ( α ) = α
α F
σ τ ( α ) = σ ( α ) = α
σ 1 ( α ) = α
E
E
F
E
E
F
E
Aut ( E )
E
F
E
F
G ( E / F ) = { σ Aut ( E ) : σ ( α ) = α for all α F }
f ( x )
F [ x ]
E
f ( x )
F
f ( x )
G ( E / F )
E
F
σ : a + b i a b i
σ ( a ) = σ ( a + 0 i ) = a 0 i = a
G ( C / R )
Q Q ( 5 ) Q ( 3 , 5 )
a , b Q ( 5 )
σ ( a + b 3 ) = a b 3
Q ( 3 , 5 )
Q ( 5 )
τ ( a + b 5 ) = a b 5
Q ( 3 , 5 )
Q ( 3 )
μ = σ τ
3
5
{ i d , σ , τ , μ }
Q ( 3 , 5 )
Q
Z 2 × Z 2
i d σ τ μ i d i d σ τ μ σ σ i d μ τ τ τ μ i d σ μ μ τ σ i d
Q ( 3 , 5 )
Q
{ 1 , 3 , 5 , 15 }
| G ( Q ( 3 , 5 ) / Q ) | = [ Q ( 3 , 5 ) : Q ) ] = 4
E
F
f ( x )
F [ x ]
G ( E / F )
f ( x )
E
f ( x ) = a 0 + a 1 x + a 2 x 2 + + a n x n
α E
f ( x )
σ G ( E / F )
0 = σ ( 0 ) = σ ( f ( α ) ) = σ ( a 0 + a 1 α + a 2 α 2 + + a n α n ) = a 0 + a 1 σ ( α ) + a 2 [ σ ( α ) ] 2 + + a n [ σ ( α ) ] n ;
σ ( α )
f ( x )
E
F
α , β E
F
Q ( 2 )
2
2
Q
x 2 2
α
β
F
σ : F ( α ) F ( β )
σ
F
f ( x )
F [ x ]
E
f ( x )
F
f ( x )
| G ( E / F ) | = [ E : F ]
[ E : F ]
[ E : F ] = 1
E = F
[ E : F ] > 1
f ( x ) = p ( x ) q ( x )
p ( x )
d
d > 1
f ( x )
F
[ E : F ] = 1
α
p ( x )
ϕ : F ( α ) E
ϕ ( α ) = β
p ( x )
ϕ : F ( α ) F ( β )
f ( x )
p ( x )
d
β E
d
ϕ : F ( α ) F ( β i )
F
β 1 , , β d
p ( x )
E ψ E F ( α ) ϕ F ( β ) F identity F
E
f ( x )
F
F ( α )
E
f ( x )
F ( β )
[ E : F ( α ) ] = [ E : F ] / d
d
ϕ
[ E : F ] / d
ψ : E E
[ E : F ]
F
σ
F
σ
F ( α )
ϕ
ϕ : F ( α ) F ( β )
F
E
[ E : F ] = k
G ( E / F )
k
p
E
F
E
F
p m
p n
n k = m
E
x p m x
p
E
x p m x
F
| G ( E / F ) | = k
G ( E / F )
G ( E / F )
σ : E E
σ ( α ) = α p n
σ
G ( E / F )
σ
Aut ( E )
α
β
E
σ ( α + β ) = ( α + β ) p n = α p n + β p n = σ ( α ) + σ ( β )
σ ( α β ) = σ ( α ) σ ( β )
σ
E
σ
G ( E / F )
F
x p n x
p
σ
F
σ
k
σ k ( α ) = α p n k = α p m = α
G ( E / F )
σ r
1 r < k
x p n r x
p m
Q ( 3 , 5 )
Q
Z 2 × Z 2
H = { i d , σ , τ , μ }
G ( Q ( 3 , 5 ) / Q )
H
G ( Q ( 3 , 5 ) / Q )
| H | = [ Q ( 3 , 5 ) : Q ] = | G ( Q ( 3 , 5 ) / Q ) | = 4
f ( x ) = x 4 + x 3 + x 2 + x + 1
Q
f ( x )
( x 1 ) f ( x ) = x 5 1
f ( x )
ω i
i = 1 , , 4
ω = cos ( 2 π / 5 ) + i sin ( 2 π / 5 )
f ( x )
Q ( ω )
σ i
Q ( ω )
σ i ( ω ) = ω i
i = 1 , , 4
G ( Q ( ω ) / Q )
[ Q ( ω ) : Q ] = | G ( Q ( ω ) / Q ) | = 4
σ i
G ( Q ( ω ) / Q )
G ( Q ( ω ) / Q ) Z 4
ω
f ( x )
F [ x ]
E
f ( x )
F [ x ]
f ( x )
E
f ( x ) = ( x α 1 ) n 1 ( x α 2 ) n 2 ( x α r ) n r = i = 1 r ( x α i ) n i
α i
f ( x )
n i
f ( x ) F [ x ]
n
n
E
f ( x )
E [ x ]
E
F
F
E
F [ x ]
f ( x )
gcd ( f ( x ) , f ( x ) ) = 1
f ( x )
F
F
0
f ( x )
F
p
f ( x )   g ( x p )
g ( x )
F [ x ]
f ( x )
char F = 0
deg f ( x ) < deg f ( x )
f ( x )
gcd ( f ( x ) , f ( x ) )   1
f ( x )
char F = p
f ( x )
f ( x )
p
f ( x ) = a 0 + a 1 x p + a 2 x 2 p + + a n x n p
F
F ( α )
E
F
α E
E = F ( α )
α
Q ( 3 , 5 ) = Q ( 3 + 5 )
Q ( 5 3 , 5 i ) = Q ( 5 6 i )
E
F
α E
E = F ( α )
F
E
F ( α , β )
f ( x )
g ( x )
α
β
K
f ( x )
g ( x )
f ( x )
α = α 1 , , α n
K
g ( x )
β = β 1 , , β m
K
1
E
F
F
a
F
a   α i α β β j
i
j
j   1
a ( β β j )   α i α
γ = α + a β
γ = α + a β   α i + a β j ;
γ a β j   α i
i , j
j   1
h ( x ) F ( γ ) [ x ]
h ( x ) = f ( γ a x )
h ( β ) = f ( α ) = 0
h ( β j )   0
j   1
h ( x )
g ( x )
F ( γ ) [ x ]
β
F ( γ )
β
g ( x )
h ( x )
β F ( γ )
α = γ a β
F ( γ )
F ( α , β ) = F ( γ )
G ( E / F )
E
F
{ σ i : i I }
F
σ i
F { σ i } = { a F : σ i ( a ) = a for all σ i }
F
σ i ( a ) = a
σ i ( b ) = b
σ i ( a ± b ) = σ i ( a ) ± σ i ( b ) = a ± b
σ i ( a b ) = σ i ( a ) σ i ( b ) = a b
a   0
σ i ( a 1 ) = [ σ i ( a ) ] 1 = a 1
σ i ( 0 ) = 0
σ i ( 1 ) = 1
σ i
F
G
Aut ( F )
G
F G = { α F : σ ( α ) = α for all σ G }
F
F { σ i }
F
{ σ i }
G
Aut ( F )
F G
σ : Q ( 3 , 5 ) Q ( 3 , 5 )
3
3
Q ( 5 )
Q ( 3 , 5 )
σ
E
F
E G ( E / F ) = F
G = G ( E / F )
F E G E
E
E G
G ( E / F ) = G ( E / E G )
| G | = [ E : E G ] = [ E : F ]
[ E G : F ] = 1
E G = F
G
E
F = E G
[ E : F ] | G |
| G | = n
n + 1
α 1 , , α n + 1
E
F
a i F
a 1 α 1 + a 2 α 2 + + a n + 1 α n + 1 = 0
σ 1 = i d , σ 2 , , σ n
G
σ 1 ( α 1 ) x 1 + σ 1 ( α 2 ) x 2 + + σ 1 ( α n + 1 ) x n + 1 = 0 σ 2 ( α 1 ) x 1 + σ 2 ( α 2 ) x 2 + + σ 2 ( α n + 1 ) x n + 1 = 0 σ n ( α 1 ) x 1 + σ n ( α 2 ) x 2 + + σ n ( α n + 1 ) x n + 1 = 0
x i = a i
i = 1 , 2 , , n + 1
σ 1
a 1 α 1 + a 2 α 2 + + a n + 1 α n + 1 = 0
a i
E
F
a i
E
F
α i
a 1
a 1 = 1
α 2 , , α n + 1
a 2
E
F
F
E
G
σ i
G
σ i ( a 2 )   a 2
σ i
G
x 1 = σ i ( a 1 ) = 1
x 2 = σ i ( a 2 )
x n + 1 = σ i ( a n + 1 )
x 1 = 1 1 = 0 x 2 = a 2 σ i ( a 2 ) x n + 1 = a n + 1 σ i ( a n + 1 )
σ i ( a 2 )   a 2
a 1 , , a n + 1 F
E
F
F [ x ]
E
E
E
F
F [ x ]
E
E [ x ]
E
F
E
F
E
F
F = E G
G
E
E
F
α
E
E = F ( α )
f ( x )
α
F
E
f ( x )
F
E
f ( x )
E
F
E G ( E / F ) = F
| G ( E / F ) | = [ E : F ]
F = E G
G
E
[ E : F ] | G |
E
F
E
F
f ( x ) F [ x ]
α
E
f ( x )
E [ x ]
G
f ( x )
E
G
α
α 1 = α , α 2 , , α n
E
g ( x ) = i = 1 n ( x α i )
g ( x )
F
g ( α ) = 0
σ
G
g ( x )
σ
g ( x )
g ( x )
g ( x )
F
deg g ( x ) deg f ( x )
f ( x )
α
f ( x ) = g ( x )
K
F
F = K G
G
K
G = G ( K / F )
F = K G
G
G ( K / F )
[ K : F ] | G | | G ( K / F ) | = [ K : F ]
G = G ( K / F )
Q ( 3 , 5 )
Q
Q
G ( Q ( 3 , 5 ) / Q )
G ( Q ( 3 , 5 ) / Q )
F
E
F
G ( E / F )
K G ( E / K )
K
E
F
G ( E / F )
F K E
[ E : K ] = | G ( E / K ) | and [ K : F ] = [ G ( E / F ) : G ( E / K ) ]
F K L E
{ i d } G ( E / L ) G ( E / K ) G ( E / F )
K
F
G ( E / K )
G ( E / F )
G ( K / F ) G ( E / F ) / G ( E / K )
G ( E / K ) = G ( E / L ) = G
K
L
G
K = L
K G ( E / K )
G
G ( E / F )
K
G
F K E
E
K
G ( E / K ) = G
K G ( E / K )
| G ( E / K ) | = [ E : K ]
| G ( E / F ) | = [ G ( E / F ) : G ( E / K ) ] | G ( E / K ) | = [ E : F ] = [ E : K ] [ K : F ]
[ K : F ] = [ G ( E / F ) : G ( E / K ) ]
K
F
σ
G ( E / F )
τ
G ( E / K )
σ 1 τ σ
G ( E / K )
σ 1 τ σ ( α ) = α
α K
f ( x )
α
F
σ ( α )
f ( x )
K
K
F
τ ( σ ( α ) ) = σ ( α )
σ 1 τ σ ( α ) = α
G ( E / K )
G ( E / F )
F = K G ( K / F )
τ G ( E / K )
σ G ( E / F )
τ ¯ G ( E / K )
τ σ = σ τ ¯
α K
τ ( σ ( α ) ) = σ ( τ ¯ ( α ) ) = σ ( α ) ;
σ ( α )
G ( E / K )
σ ¯
σ
K
σ ¯
K
F
σ ( α ) K
α K
σ ¯ G ( K / F )
G ( K / F )
F
β
K
G ( K / F )
σ ¯ ( β ) = β
σ G ( E / F )
β
F
G ( E / F )
K
F
G ( K / F ) G ( E / F ) / G ( E / K )
σ G ( E / F )
σ K
K
σ
K
K
σ K G ( K / F )
ϕ : G ( E / F ) G ( K / F )
σ σ K
ϕ ( σ τ ) = ( σ τ ) K = σ K τ K = ϕ ( σ ) ϕ ( τ )
ϕ
G ( E / K )
| G ( E / F ) | / | G ( E / K ) | = [ K : F ] = | G ( K / F ) |
ϕ
G ( K / F )
ϕ
G ( K / F ) G ( E / F ) / G ( E / K )
G ( E / F )
E
E { id } L G ( E / L ) K G ( E / K ) F G ( E / F )
f ( x ) = x 4 2
Q
x 4 2
f ( x )
Q ( 2 4 , i )
f ( x )
( x 2 + 2 ) ( x 2 2 )
f ( x )
± 2 4
± 2 4 i
2 4
Q
i
x 2 + 1
Q ( 2 4 )
f ( x )
Q ( 2 4 ) ( i ) = Q ( 2 4 , i )
[ Q ( 2 4 ) : Q ] = 4
i
Q ( 2 4 )
[ Q ( 2 4 , i ) : Q ( 2 4 ) ] = 2
[ Q ( 2 4 , i ) : Q ] = 8
{ 1 , 2 4 , ( 2 4 ) 2 , ( 2 4 ) 3 , i , i 2 4 , i ( 2 4 ) 2 , i ( 2 4 ) 3 }
Q ( 2 4 , i )
Q
Q
Q ( 2 4 , i )
G
f ( x )
8
σ
σ ( 2 4 ) = i 2 4
σ ( i ) = i
τ
τ ( i ) = i
G
4
2
G
{ i d , σ , σ 2 , σ 3 , τ , σ τ , σ 2 τ , σ 3 τ }
τ 2 = i d
σ 4 = i d
τ σ τ = σ 1
G
D 4
G
x 4 2
f ( x )
f ( x )
n
a x 2 + b x + c = 0
x = b ± b 2 4 a c 2 a
n
E
F
F = F 0 F 1 F 2 F r = E
i = 1 , 2 , , r
F i = F i 1 ( α i )
α i n i F i 1
n i
f ( x )
F
K
f ( x )
F
F
f ( x )
f ( x )
x n a
x n 1
n
x n 1
n
n
x n 1
Q
1 , ω , ω 2 , , ω n 1
ω = cos ( 2 π n ) + i sin ( 2 π n )
x n 1
Q
Q ( ω )
G
G = H n H n 1 H 1 H 0 = { e }
H i
H i + 1
G
{ H i }
H i + 1 / H i
{ i d } A 3 S 3
S 3
S 5
F
E
x n a
F
a F
G ( E / F )
x n a
a n , ω a n , , ω n 1 a n
ω
n
F
n
ζ
x n a
x n a
ζ , ω ζ , , ω n 1 ζ
E = F ( ζ )
G ( E / F )
x n a
G ( E / F )
σ
τ
G ( E / F )
σ ( ζ ) = ω i ζ
τ ( ζ ) = ω j ζ
F
σ τ ( ζ ) = σ ( ω j ζ ) = ω j σ ( ζ ) = ω i + j ζ = ω i τ ( ζ ) = τ ( ω i ζ ) = τ σ ( ζ )
σ τ = τ σ
G ( E / F )
G ( E / F )
F
n
ω
n
α
x n a
α
ω α
x n a
ω = ( ω α ) / α
E
K = F ( ω )
F K E
K
x n 1
K
F
σ
G ( F ( ω ) / F )
σ ( ω )
σ ( ω ) = ω i
i
x n 1
ω
τ ( ω ) = ω j
G ( F ( ω ) / F )
σ τ ( ω ) = σ ( ω j ) = [ σ ( ω ) ] j = ω i j = [ τ ( ω ) ] i = τ ( ω i ) = τ σ ( ω )
G ( F ( ω ) / F )
{ i d } G ( E / F ( ω ) ) G ( E / F )
G ( E / F ( ω ) )
G ( E / F ) / G ( E / F ( ω ) ) G ( F ( ω ) / F )
G ( E / F )
F
F = F 0 F 1 F 2 F r = E
F
F = K 0 K 1 K 2 K r = K
K
E
K i
K i 1
E
F
F = F 0 F 1 F 2 F r = E
i = 1 , 2 , , r
F i = F i 1 ( α i )
α i n i F i 1
n i
F
F = K 0 K 1 K 2 K r = K
K E
K 1
x n 1 α 1 n 1
α 1 , α 1 ω , α 1 ω 2 , , α 1 ω n 1 1
ω
n 1
F
n 1
K 1 = F ( α ! )
F
n 1
β
x n 1 α 1 n 1
x n 1 α 1 n 1
β , ω β , , ω n 1 1
ω
n 1
K 1 = F ( ω β )
K 1
F
F 1
F = K 0 K 1 K 2 K r = K
K i
K i 1
K i F i
i = 1 , 2 , , r
f ( x )
F [ x ]
char F = 0
f ( x )
f ( x )
F
f ( x )
E
F
F = F 0 F 1 F n = E
E
f ( x )
F i
F i 1
G ( E / F i )
G ( E / F i 1 )
G ( E / F )
{ i d } G ( E / F n 1 ) G ( E / F 1 ) G ( E / F )
G ( E / F i 1 ) / G ( E / F i ) G ( F i / F i 1 )
G ( F i / F i 1 )
G ( E / F )
S 5
p
S p
p
S p
G
S p
σ
τ
p
σ = ( 1 2 )
τ
p
τ n
p
1 n < p
μ = τ n = ( 1 2 i 3 i p )
n
1 n < p
( 1 2 ) ( 12 i 3 i p ) = ( 2 i 3 i p )
( 2 i 3 i p ) k ( 12 ) ( 2 i 3 i p ) k = ( 1 i k )
( 1 n )
1 n < p
S p
( 1 j ) ( 1 i ) ( 1 j ) = ( i j )
S p
f ( x ) = x 5 6 x 3 27 x 3
f ( x ) = x 5 6 x 3 27 x 3 Q [ x ]
f ( x )
Q
S 5
f ( x )
f ( x )
f ( x ) = 5 x 4 18 x 2 27
f ( x ) = 0
f ( x )
x = ± 6 6 + 9 5
f ( x )
f ( x )
3
2
2
0
0
4
f ( x )
f ( x )
K
f ( x )
f ( x )
K
K
Q
f ( x )
G ( K / Q )
S 5
f
σ G ( K / Q )
σ ( a ) = b
a
b
f ( x )
C
a + b i a b i
G ( K / Q )
α
f ( x )
[ Q ( α ) : Q ] = 5
Q ( α )
K
[ K : Q ]
[ K : Q ] = | G ( K / Q ) |
G ( K / Q ) S 5
G ( K / Q )
5
S 5
5
G ( K / Q )
S 5
S 5
f ( x )
C [ x ]
C
E
α E
E = C ( α )
α
f ( x )
C [ x ]
L
f ( x )
C
E
L
C
L
C
L
f ( x ) ( x 2 + 1 )
R
L
R
K
G
G ( L / R )
L K R
| G ( L / K ) | = [ L : K ]
[ L : R ] = [ L : K ] [ K : R ]
[ K : R ]
K = R ( β )
β
f ( x )
K = R
G ( L / R )
G ( L / C )
2
L   C
| G ( L / C ) | 2
G
G ( L / C )
E
G
[ E : C ] = 2
γ E
x 2 + b x + c
C [ x ]
( b ± b 2 4 c ) / 2
C
b 2 4 c
C
L = C
Q
G ( Q ( 30 ) / Q )
G ( Q ( 5 4 ) / Q )
G ( Q ( 2 , 3 , 5 ) / Q )
G ( Q ( 2 , 2 3 , i ) / Q )
G ( Q ( 6 , i ) / Q )
Z 2
Z 2 × Z 2 × Z 2
x 3 + 2 x 2 x 2
Q
x 4 + 2 x 2 + 1
Q
x 4 + x 2 + 1
Z 3
x 3 + x 2 + 1
Z 2
Q
x 3 + 2 x 2 x 2 = ( x 1 ) ( x + 1 ) ( x + 2 )
Z 3
x 4 + x 2 + 1 = ( x + 1 ) 2 ( x + 2 ) 2
GF ( 729 )
GF ( 9 )
[ GF ( 729 ) : GF ( 9 ) ] = [ GF ( 729 ) : GF ( 3 ) ] / [ GF ( 9 ) : GF ( 3 ) ] = 6 / 2 = 3
G ( GF ( 729 ) / GF ( 9 ) ) Z 3
G ( GF ( 729 ) / GF ( 9 ) )
σ
σ 3 6 ( α ) = α 3 6 = α 729
α GF ( 729 )
Q [ x ]
x 5 12 x 2 + 2
x 5 4 x 4 + 2 x + 2
x 3 5
x 4 x 2 6
x 5 + 1
( x 2 2 ) ( x 2 + 2 )
x 8 1
x 8 + 1
x 4 3 x 2 10
S 5
S 3
Q [ x ]
x 4 1
x 4 8 x 2 + 15
x 4 2 x 2 15
x 3 2
Q ( i )
Z 2
S 3
Z 3
E
F [ x ]
[ E : F ]
6
3
G ( E / F )
S 3
3
Z 3
S 3
F K E
E
F
E
K
G
n
| G |
n !
G
S n
F E
f ( x )
F
f ( x )
E
f ( x )
Q [ x ]
7
p
f ( x ) Q [ x ]
p
S p
p
p 5
p
p
Z p ( t )
Z p
f ( x ) = x p t
Z p ( t ) [ x ]
f ( x )
E
F
K
L
σ G ( E / F )
σ ( K ) = L
K
L
K
L
G ( E / K )
G ( E / L )
G ( E / F )
σ Aut ( R )
a
σ ( a ) > 0
K
x 3 + x 2 + 1 Z 2 [ x ]
K
F
char ( F )   2
f ( x ) = a x 2 + b x + c
F ( α )
α = b 2 4 a c
K
F
E
F
K
[ E : F ] = 2
E
F [ x ]
Φ p ( x ) = x p 1 x 1 = x p 1 + x p 2 + + x + 1
Q
p
ω
Φ p ( x )
Q ( ω )
ω , ω 2 , , ω p 1
Φ p ( x )
Φ p ( x )
G ( Q ( ω ) / Q )
p 1
G ( Q ( ω ) / Q )
Q
ω , ω 2 , , ω p 1
ω   1
ω i
Φ p
Φ p ( ω i )
ω
ω , ω 2 , , ω p 1
ϕ i : Q ( ω ) Q ( ω i )
ϕ i ( a 0 + a 1 ω + + a p 2 ω p 2 ) = a 0 + a 1 ω i + + c p 2 ( ω i ) p 2
a i Q
ϕ i
ϕ 2
G ( Q ( ω ) / Q )
{ ω , ω 2 , , ω p 1 }
Q ( ω )
Q
ω , ω 2 , , ω p 1
G ( Q ( ω ) / Q )
F
E
F
G ( E / F )
F K L E
{ i d } G ( E / L ) G ( E / K ) G ( E / F )
F
f ( x ) F [ x ]
n
E
f ( x )
α 1 , , α n
f ( x )
E
Δ = i < j ( α i α j )
f ( x )
Δ 2
f ( x ) = x 2 + b x + c
Δ 2 = b 2 4 c
f ( x ) = x 3 + p x + q
Δ 2 = 4 p 3 27 q 2
Δ 2
F
σ G ( E / F )
f ( x )
σ ( Δ ) = Δ
σ G ( E / F )
f ( x )
σ ( Δ ) = Δ
G ( E / F )
A n
Δ F
x 3 + 2 x 4
x 3 + x 3
p ( x ) = x 4 2
8
8
8
8
p ( x ) = x 4 2
2 1 4 = 2 4
2 1 4 i = 2 4 i
S 4
8
8
τ ( x ) = τ ( i = 0 7 q i c i ) q i Q = i = 0 7 τ ( q i ) τ ( c i ) τ is a field automorphism = i = 0 7 q i τ ( c i ) rationals are fixed
τ
τ ( c )
τ ( c k ) = ( τ ( c ) ) k
2
1
2 4 i
2
2 4 i
Q ( 2 4 i )
Q ( 2 4 i )
2 4 i 2 4 = ( 1 i ) 2 4
Q ( 2 4 i 2 4 ) = Q ( ( 1 i ) 2 4 )
x 4 + 8
( 1 i ) 2 4
x 4 + 8
2
2
2
Z 2 × Z 2
2
Q ( 2 )
14
2 i
i
Q ( i )
4
2
Q ( 2 4 )
x 4 2
Q ( 2 4 )
H = ( 1 , 4 )
Q ( 2 4 )
x 4 2
2
4
p ( x ) = x 3 6 x 2 + 12 x 10
p ( x )
x 5 x 1
S 5
S 5
p ( x ) = x 4 + x + 1
p ( x )
p ( x )
Q
p ( x )
Q
x 5 x 1
3 6
x 6 + x 2 + 2 x + 1
p ( x ) = x 7 7 x + 3
y 2 = x ( 81 x 5 + 396 x 4 + 738 x 3 + 660 x 2 + 269 x + 48 )
p ( x )
P S L ( 2 , 7 )
S L ( 2 , 7 )
2 × 2
Z 7
P S L ( 2 , 7 )
{ I 2 , I 2 }
A 5
7
6
168